Tuan Minh Pham
email: tuan-minh.pham@sophia.inria.fr

Yves Bertot
email: yves.bertot@inria.fr

A combination of a dynamic geometry software with a proof assistant for interactive formal proofs

Keywords: interactive geometric proofs, Coq, proof assistant, dynamic geometry

. Thanks to the features of Geogebra, users can create and manipulate geometric constructions, they discover conjectures and interactively build formal proofs with the support of Coq. Our system allows users to construct fully traditional proofs in the same style as the ones in high school. For each step of proving, we provide a set of applicable rules verified in Coq for users, we also provide tactics in Coq by which minor steps of reasoning are solved automatically.

Introduction

Nowadays, dynamic geometry software (DGS) plays an important role and has highly influenced mathematics education. Students can use it to construct geometric objects, observe their changes by moving free points or applying euclidian transformations, then they can discover conjectures. There are numerous softwares for interactive geometry on the market with a variety of features, many of them are really used in classroom in many countries.

However, these uses are usually limited at the level of discovering conjectures. Some DGS provide proof feature by combining with automated geometry theorem proving (GTP), allow users to verify conjectures. They relies on several efficient automatic proof methods, such as Gröbner bases method [START_REF] Kapur | Using Gröbner Bases to reason about geometry problems[END_REF], Wu's method [START_REF] Chou | An introduction to Wu's method for mechanical theorem proving in geometry[END_REF], area method [START_REF] Chou | Machine proofs in geometry: Automated production of readable proofs for Geometry Theorems[END_REF] and full-angles method [START_REF] Chou | Automated generation of readable proofs with geometric invariants. Theorem proving with full-angles[END_REF]. The first two ones are algebraic methods which use polynomials to solve problems, they do not give us human-readable proofs. The last two ones can produce human-readable proofs, nevertheless these proofs are not fully traditional proofs, they do not look like high school proofs.

We aim at constructing a tool for students to prove geometric theorems interactively. Students can effectuate reasoning step-by-step as they learn in the high school curriculum which can use backward/forward proof method and auxiliary lines as well. So here, we present GeoCoq, an interface for interactive geometry proving which is a combination of Geogebra with Coq. Reasoning steps are built by mouse clicks, and are sent to Coq. For its part, Coq executes the reasoning steps, the response is sent back to the user to continue their proof.

With the support of a proof assistant like Coq, reasoning steps are verified, by which the correctness of proofs is guaranteed. This is an advantage in using Coq because geometry reasoning usually uses tacit assumptions based on visual evidence without verification.

Another advantage relies on the Coq system's language Ltac, with which we can develop automatic tactics which allow users to automatically solve minor steps of reasoning. Users do not have to delve into details of proofs which lead to technical proofs and are not suitable with their thought. In addition, with the formalization of automatic proof methods as tactics in Coq (such as the area method which is formalized by J.Narboux [START_REF] Narboux | A Decision Procedure for Geometry in Coq. of Theorem Proving in Higher-Order Logics[END_REF]), users can use these methods to check the fact before proving it, or alternatively use them in steps of constructing proofs.

This paper is organized as follows. In section 2, we give a short description about Geogebra. The Coq system, the communication using Pcoq, and a brief description about the library for geometry are described in section 3. Section 4 presents the interface and provided features for proofs. The next section deals with our implementations. We mention some related work in section 6 and the last section is for our conclusions and perspectives.

Geogebra

Geogebra is a free dynamic geometry software for education in schools. It was started by Markus Hohenwarter in 2001 in his master and PhD work. It is implemented in Java and thus available for multiple platforms. It provides a new kind of tool for mathematics by joining geometry, algebra and calculus.

It received several international educational software awards and is applied in education at schools in different countries. Like other dynamic geometry systems, Geogebra provides basic geometric objects such as points, vectors, segments, straight lines, circle. . . , other geometric constructions which are more complex such as midpoint, parallel lines, circumcircle. . . Users can construct geometric objects and manipulate them. Geogebra also allows students to undo or redo constructions at anytime. By moving free points in figures students can find out conjectures and check theirs correctness, checks are limited to simple relations of 2 objects at the moment.

A strong point of Geogebra which makes it different from other DGSs is the connection of Dynamic Geometry and Computer Algebra System (CAS). Geogebra can profit both from visualization capabilities of CAS and dynamic changeability of DGS. It encourages students to discover mathematics in a bidirectional experimental way. Students can investigate equations corresponding to drawn objects and they can also investigate the figures corresponding togiven equations.

By combining Geogebra and Coq, we would like to provide an interface which allows students to have many points of view on a geometric problem.

Coq communication and Geometry library

Coq is a proof development system, which allows users to define functions or predicates, state mathematical theorems and interactively develop formal proofs. Coq is not an automated theorem prover, however it provides a tactic language letting users define their own proof method and it also includes tactics for automatic theorem proving and decision procedures. So Coq is adapted to our needs in interactively constructing proofs. Some integrated development environments (IDE) can be used to communicate with Coq. Here, we are interested with Pcoq which is a graphical user-interface for Coq. Using Pcoq to communicate with Coq offers some benefits. Pcoq is implemented in Java, the same programming language as Geogebra, so this makes it easy to be integrated in Geogebra. Pcoq manipulates all formulas and commands as tree-like structures (also known as abstract syntax trees) rather than plain text. This makes it easy to attach special mathematical notations to some functions, and to attach sentences in natural language to geometrical statements. Furthermore, this provides an easy access to the structure of information (hypotheses, goals) that we receive from Coq.

In order to be able to reason in Coq, we still need a geometry library, in which geometry basic notions and their properties are formalized, geometry rules that we want to provide for users are verified. An interesting formalization was developed by F.Guilhot for the French high school curriculum [START_REF] Guilhot | Formalisation en Coq et visualisation d'un cours de géométrie pour le lycée[END_REF]. It is based on providing a new axiom system that is more adapted to the knowledge of students. It covers a large portion of basic notions of plane geometry, properties and theorems. Moreover, proofs are traditional, they are formalized in the manner of reasoning suitable to the capacity of high school students. Among the classic theorems which are proved in this library, we can cite Menelaus, Ceva, Desargues, Pythagoras, Simson's line. . .

The interface

Our interface is a combination of Geogebra with Pcoq. We integrate Pcoq as a view of Geogebra (figure 1). Like other views, it can be shown, hidden, dragged around to modify its position, and opened as an external window.

Two main sub windows of Pcoq are the command window and proof window. The command window is used to display all Coq commands which state and prove theorems. The proof window allows users to construct proofs, they can see hypotheses, and goals need to be proved for each step of geometry reasoning in this window. In addition, other sub windows are used to display applicable rules for a goal, to apply rules by giving values for their parameters, and to announce errors. . . We detail the interface and its features by showing how users can state and prove theorems. The following theorem will be used as an example for the rest of this section:

Constructive form

Predicate form

M = midPoint (A, B) collinear A B M ∧ MA = MB M = intersectPoint (l 1 , l 2 ) l 1 ∦ l 2 ∧ M ∈ l 1 ∧ M ∈ l 2 l = line (A, B) A = B ∧ A ∈ l ∧ B ∈ l l = parallelLine (A, m) A ∈ l ∧ l m l = perpendicularLine (A, m) A ∈ l ∧ l ⊥ m 4.1 Stating theorems 4.1.

Constructing diagrams

Geogebra provides common geometry construction tools that allow users to construct diagrams. Geometry objects are created one by one by using appropriate construction tools and selecting existent objects. Users can move free points, and dependent constructions will be updated, this can help users discover conjectures.

Here, to construct a diagram corresponding to the above example, users draw a triangle ABC, construct perpendicular lines d ⊥ AC and e ⊥ AB, take intersection points E and D, they complete by taking midpoints G and F of BC and DE respectively and joining them.

Each construction is translated into commands in Coq (figure 1). Adding new geometry objects or removing existent geometry objects implies adding or removing corresponding hypotheses. In most geometry theorem provers, a predicate form is used to describe geometric statement, i.e. hypotheses and conclusion are represented by geometry predicates. Our tool is not an exception, however in the phase of stating theorem, we have geometric statement in constructive form (left column of table 1). In theorem proving phase which will be presented in the next section, users can discover corresponding geometric predicates (right column of table 1) using our tactics in Coq to unroll definitions in hypotheses.

Discovering conjectures

Once a diagram is completely constructed, users easily see free points that do not depend on other constructions. Users can move these free points, dependent constructions are updated, and users can observe relationships between geometry objects, traces of points. . . If they see that a conjecture seems to be true, they can decide to prove it. We can pre-verify the correctness of conjecture to ensure that users are going to prove a fact. This pre-verification can Fig. 2. Users find and add a conclusion be performed using Geogebra feature with the support of its CAS, or using automatic proof method in Coq (such as the area method). For our example, users can move points A, B and C, they find out perpendicularity of GF and DE (GF ⊥ DE). They select these lines and right click, a dialog appears to confirm this perpendicularity, and ask users to prove it (figure 2). Once users decide to prove the conjecture, we can go to theorem proving phase in the next section.

Theorem proving

We start this section by discovering geometric predicates from the construction. Our implementation allows users to get geometric predicates automatically or manually. We find definitions of geometric objects in hypotheses, and use our tactics in Coq to get description of these definitions. Ltac unroll AllDef inition := match goal with . . . |H :?M = intersectionP oint?a?b ⇒ let(H1H2H3) := f resh "intersectionP ointP roperty" in destruct (@unroll intersectionP oint M a b) as [H1[H2H3]]; auto; revert H; clear H; unroll AllDef inition; intro H . . . end.

When applying this tactic, the system searches in the hypotheses to find definitions in the first column of table 1 and gives the corresponding predicates in the second column.

This form contains predicates such as l 1 ∦ l 2 , A = B. . . which may be used as non degeneracy conditions for the existence of objects. With many definitions of geometric objects, we implicitly add these non degeneracy conditions in hypotheses. These conditions are very important for reasoning, sometimes resolving a problem in degenerate cases is more complex than resolving the original problem. The same diagram with differents way of construction will have different non degeneracy conditions.

Searching and Applying rules

Here, we present backward-chaining approach for geometry reasoning. This approach progresses from the conclusion to the hypotheses, i.e. we need to prove ∀GeometricElements, Hyp 1 ∧ . . . ∧ Hyp n → Goal. We search in a rule set to find a rule of the following form ∀GeometricElements, G 1 ∧ . . . ∧ G n → Goal. The problem evolves into proving the subgoals G 1 , . . . , G n . This process is repeated for each subgoal until the subgoal is in the hypotheses or is an axiom.

Searching applicable rules in our interface is implemented using Search tactics. It is strong enough to find out all rules in a database which lead to a goal given by a pattern. From proof window of the interface, users can search all applicable rules for the current goal. A list of applicable rules will be displayed, allowing users to select an appropriate rule. After giving values for parameters of this rule which usually are points, lines. . . , users can update statement of the rule with given values, the system checks correctness by verifying its hypotheses.

Returning to our example 4.1, to prove that GF ⊥ DE, we need to find rules which have conclusion in the form of perpendicularity of 2 lines. A variant of Search tactics is sent to Coq SearchP attern ( ⊥ ) inside M oduleN ame With a list of applicable rules (the upper figure of figure 3), we can select the following rule to apply.

Lemma isosceles perp :

∀ A B C M : P oint, B = C → A = M → AB = AC → M = M idP oint(B, C) → AM ⊥ BC.
Using the four points G, D, E and F as values for its parameter respectively, we have an instance of this rule (the lower figure of figure 3)

D = E → G = F → GD = GE → F = M idP oint(D, E) → GF ⊥ DE.
By construction of F, we have F = M idP oint(D, E), so applying this rule is reasonable, it is performed by tactics apply isosceles perp with (A:=G)(B:=D)(C:=E)(M:=F). Now, we have to prove that GD = GE in place of proving that GF ⊥ DE.

Adding new properties

Generally, we can not use only backward reasoning to solve problems, an alternative way is to use forward-chaining. This process aims at generating new properties from the hypotheses by applying given rules. These properties are used as hypotheses in the next time of the process. The process will finish when it can generate the goal. If we need to prove that ∀GeometricElements, Hyp 1 ∧. . .∧Hyp n → Goal. And in the base of rules we have ∀GeometricElements, H j 1 ∧. . .∧H jm → Goal j , with {H j 1 , . . . , H jm } are a subset of {Hyp 1 , . . . , Hyp n }, so we can add G j as a hypothesis of theorem.

Our system provides a variant of forward-chaining, allowing users to add new properties while proving theorems. A property is used as an hypothesis if it is proved. In our example, to prove that GD = GE, users try to prove that GD = GC and GE = GC. For the first, users select 2 segments GD and GC, with a right click users can find out their equal relation and add this equation to the proof. To prove GD = GC, we will use the following rule:

Lemma rightT riangle midP oint :

∀A B C M : P oint, A = B → A = C → line(A, B) ⊥ line(A, C) → M = M idP oint(B, C) → M A = M B ∧ M A = M C
With the configuration of 4 points D, B, C and G, we have:

A = B → A = C → line(A, B) ⊥ line(A, C) → M = M idP oint(B, C) → M A = M B ∧ M A = M C.
It is the same way for the second one GE = GC. It remains to prove that line(B, D) ⊥ line(C, D) and line(C, E) ⊥ line(B, E). They will be proved thanks to the construction of points D, E and using automatic tactics that we will present later.

Drawing auxiliary lines

Sometime, users need to add lines, ray, segment. . . to a diagram during their proofs. It is quite important to be sure the existence of these objects. In theorem stating phase, each created object maybe given us some conditions which relate to its existence. However, in theorem proving phase, to create a new object, we have to verify all conditions to guarantee its existence. Depending on which object users need to create in this phase, corresponding commands will be sent to Coq. The following commands are for adding the intersection point M of 2 lines l 1 and l 2 . letH := f resh "isIntersectionP oint" in assert(H : existsM, M = IntersectionP oint l 1 l 2 ). apply exists intersectionP oint. These commands are sent to Coq, the system requests to prove l 1 ∦ l 2 for existence of M. Once this condition is proved, the point M will be displayed in drawing window, and the command destruct H as [M, H]. automatically is sent to Coq to have M = IntersectionP oint l 1 l 2 ) in hypotheses.

Automatic tactics

Proving geometry with the support of Coq gives a high level of confidence, each reasoning step is verified. However, it also complicates the process of proving. Users not only decide which rule will be applied but also prove many minor goals which lead to tedious proofs and is not suitable with a pedagogical setting. So, to avoid drowning users into details of proof, we try to provide tactics to automatically solve these problems.

In this context, we present here a tactic to solve problems of line and points on the line. In So, our tactics is a combination of "unroll AllDefinition" and "replaceAl-lLineInstances" tactics.

Ltac replace auto Lines := solve [unroll AllDef inition; replaceAllLineInstances].

For each particular goal, corresponding tactics will be provided in a popup menu (figure 4).

Implementation

Geogebra is implemented in Java, its architecture is clear and organized in separate layers and modules. In the last version (v3.3.69), each view of Geogebra (such as drawing pad, algebra view, CAS view, Spreadsheet view) is independently implemented by extending a JPanel class of Java and implementing a View interface. Hence, some features are provided that allow users to show or hide iews, drag around to modify their position, and open them in an external window as well.

public interf ace V iew{ publicvoidadd (GeoElementgeo); publicvoidremove (GeoElementgeo); publicvoidupdate (GeoElementgeo); } These views work in synchronized mode thanks to a controller module. This notifies every change of geometric objects to all system view by calling corresponding functions in each view interface. set of basic rule, and allows to find out traditional proofs. GeoExpert allows users to visually understand each step in generated proof.

Geometry Explorer [START_REF] Wilson | Combining Dynamic Geometry, Automated Geometry Theorem Proving And Diagrammatic Proofs[END_REF] also implements the full-angles method. However, it can automatically generate novel diagrammatic proofs corresponding with reasoning used by its GTP. It is a good illustration for proofs.

Geoproof [START_REF] Narboux | A Graphical User Interface for Formal Proofs in Geometry[END_REF] is very close to our system. Reasoning in Geoproof is performed by Coq and based on formalization of the area method in this proof assistant [START_REF] Narboux | A Decision Procedure for Geometry in Coq. of Theorem Proving in Higher-Order Logics[END_REF]. Since Geoproof also uses Coq, so we will have some similar points in stating phase. The main differences lie on theorem proving phase and the way the system connects to Coq. Geoproof only allows users to construct theorem statements, send them to a CoqIDE and check this fact by apply the area method. It does not allow to construct traditional proofs and provide interactions between DGS and Coq.

Another one which combines a DGS with Coq is GeoView [START_REF] Bertot | Visualizing Geometrical Statements with GeoView[END_REF], but this tool works in the opposite direction. It produces a diagram in a DGS namely GeoPlan from a theorem statement in Coq.

Conclusion and Future Work

In this paper, we present our interface which allows users to interactively construct traditional proofs in Geometry. Reasoning methods at high school level such as forward method, backward method, and drawing auxiliary lines as well are provided. Steps of reasoning are verified by a proof assistant hence the correctness is guaranteed.

Integrating a proof assistant in a dynamic geometry software offers a novel way in learning geometry. It allows users to additionally have a logic view in solving geometric problems. Users know what they have in hypotheses, what they have to prove, which rules they can apply. . . By which, they can take good decisions and thoroughly understand reasoning steps.

Some features need to be improved to allow users to easily use our tool. Dynamically constructing diagram from statement of rules is one of them. With constructed diagram of a rule, users easily find points, lines that their configuration conforms with this diagram. Then users can drag and drop these objects to apply this rule. The second one is how we can organize and manage the set of applicable rules and the set of tactics while adapting to user levels.

Fig. 1 .

 1 Fig. 1. A screen-shot of GeoCoq GUI. Pcoq is integrated as the windows in the left

Example 4 . 1

 41 Let BD and CE be two altitudes of triangle ABC and points G and F be the midpoints of BC and DE respectively. It holds that GF ⊥ DE.

Fig. 3 .

 3 Fig. 3. Applicable rules are displayed at the bottom, users select values to apply the rule

  our example, we have to prove that line(B, D) ⊥ line(C, D) while we have c = line(A, C), d = perpendicularLine(B, c), D = IntersectionP oint(c, d). With the "unroll AllDefinition" tactics presented above, we will get A ∈ c ∧ C ∈ c, B ∈ d ∧ d ⊥ c and D ∈ c ∧ D ∈ d. The following tactics try to replace all instance of line(M, N ) by line l if it finds M ∈ l and N ∈ l in hypotheses. Ltac replaceAllLineInstances := match goal with context[line ?M ?N ] ⇒ match goal with |H1 : liesOnLine M ?l ⇒ match goal with |H2 : liesOnLine M l ⇒ try (replace (line(M, N )) with l in * ; auto with GeoBase); replaceAllLineInstances end

Fig. 4 .

 4 Fig. 4. Users can use automatic tactics in popup menu

Table 1

 1 Constructive and predicate form of some constructions

Our integration of Pcoq in Geogebra respects this architecture (figure 5. Pcoq is implemented as a view of Geogebra. Once it receives a notification, its information will be treated, corresponding commands will be produced and sent to Coq. As we said, communicated information are in tree format, so a module to produce tree format commands from a construction is necessary. The following code lines is an example for adding a midpoint in theorem stating phase: public T ree[] add(GeoElement geo){ //this f unction is called when a object is created . . . if (parentAlgo instanceof AlgoM idpoint){ //midpoint of a segment contents = commandM idP oint(geo, proof M ode); } return contents; } private T ree[] commandM idP oint(GeoElement geo, boolean proof M ode){ //this f unction is to produced commands f or a midpoint T ree[] contents = null; . . . AlgoElementparentAlgo = geo.getP arentAlgorithm(); GeoElementpoint1 = ((AlgoM idpoint)parentAlgo).getInput()[0]; GeoElementpoint2 = ((AlgoM idpoint)parentAlgo).getInput() [START_REF] Amerkad | Mathematics and Proof Presentation in Pcoq. Proof Transformation and Presentation and Proof Complexities in connection with IJCAR[END_REF]; contents = new T ree [START_REF] Bertot | Visualizing Geometrical Statements with GeoView[END_REF]; contents[0] = T reeF ormat.addV ar(geo.getLabel(), T reeF ormat.P oint); contents [START_REF] Amerkad | Mathematics and Proof Presentation in Pcoq. Proof Transformation and Presentation and Proof Complexities in connection with IJCAR[END_REF] = T reeF ormat.addHypothesis("Hyp M idP oint", T reeF ormat.assignV ariable(geo.getLabel(), T reeF ormat.f unction(T reeF ormat.M idP ointAB, point1.getLabel(), point2.getLabel()))); return contents; } In our implementation, we continue the notion of "proof by pointing" [START_REF] Bertot | Proof by pointing[END_REF] which has been already approached in Pcoq. It allows users to guide precisely the proof process using the mouse in the user-interface of proof assistants i.e., the gestures of pointing at a subexpression of goals or assumptions is enough to synthesize appropriate commands for the system.

While proof-by-pointing in Pcoq relies on analyzing formulas and understanding the meaning of logical symbols, this notion in our system is realized by the geometric signification of formulas. For each reasoning step, by finding out signification of subgoal, hypotheses that users selected, the system can provide appropriate tactics in a popup menu to guide users to solve the problem. The following code lines determine if the selected goal has the format of ( = line ), ( = perpendicularline ) or ( = parallelLine ), in this case the system provides replace auto Lines tactics to solve it.

. . . T ree lef tEqual = selectedItem.sons() [START_REF] Amerkad | Mathematics and Proof Presentation in Pcoq. Proof Transformation and Presentation and Proof Complexities in connection with IJCAR[END_REF].sons()[0]; T ree rightEqual = selectedItem.sons() [START_REF] Amerkad | Mathematics and Proof Presentation in Pcoq. Proof Transformation and Presentation and Proof Complexities in connection with IJCAR[END_REF].sons() [START_REF] Amerkad | Mathematics and Proof Presentation in Pcoq. Proof Transformation and Presentation and Proof Complexities in connection with IJCAR[END_REF]; if (lef tEqual.operatorN ame().equalsIgnoreCase("appc")){ String f uncN ame = lef tEqual.sons()[0].atomV alue(); //verif y conditions to add a menu item in popup menu if (f uncN ame.equals(LineAB) f uncN ame.equals(LineOAb) f uncN ame.equals(LineP Ab)){ miT mp = newM yM enuItem(autoLineT acticsAction, autoLineT acticsN ame, menuT ype); miList.add(miT mp); }} . . .

Related Work

DGSs are more and more used in education. Many works have been performed to provide combinations of DGS and GTP. Some of them use automatic methods, allow users to prove geometry theorems. We will cite here several systems that can generate readable proofs.

Geothms [START_REF] Quaresma | GeoThms -a Web System for Euclidean Constructive Geometry[END_REF] with a web interface uses the GCLC prover which is based on the area method. With the support of repository of geometry theorems, users can store theorem statements and their proofs.

MMP/Geometer [START_REF] Gao | MMP/Geometer -a software package for automated geometry reasoning[END_REF] and GeoExpert [START_REF] Gao | [END_REF] are strong systems which implement Wu's method, the area method, the full-angles method (for GeoExpert), and especially the deductive database method [START_REF] Chou | A deductive database approach to automated geometry theorem proving and discovering[END_REF]. The last method relies on a


Xuan Le

Truong

Thi Phuong Le

Ngoc

Alain Pham
email: alain.pham@univ-orleans.fr

Ngoc Dinh

Thanh Long Nguyen

Long
email: longnt@hcmc.netnam.vn

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

Keywords: Nonlinear wave equation, local existence, global existence, blow up, exponential decay AMS subject classification: 35L05, 35L15, 35L70, 37B25

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.

Introduction

In this paper, we consider the following nonlinear wave equation with initial conditions and boundary conditions of two-point type

u tt -u xx + u + λu t = |u| p-2 u, 0 < x < 1, t > 0, (1.1)
u x (0, t) = -|u(0, t)| α-2 u(0, t) + λ 0 u t (0, t) + h 1 (t)u(1, t) + λ 1 u t (1, t), t > 0, (

-u x (1, t) = -|u(1, t)| β-2 u(1, t) + λ 1 u t (1, t) + h 0 (t)u(0, t) + λ 0 u t (0, t), t > 0, (1.3) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), (1.4) where λ 0 , λ 1 , λ 0 , λ 1 , λ, p are constants and u 0 , u 1 , h 0 , h 1 are given functions satisfying conditions specified later.

The wave equation u tt -∆u = f (x, t, u, u t ), (1.5) with the different boundary conditions, has been extensively studied by many authors, see ( [START_REF] Bergounioux | Mathematical model for a shock problem involving a linear viscoelastic bar[END_REF], [START_REF] Clark | Global classical solutions to the Cauchy problem for a nonlinear wave[END_REF], [START_REF] Long | On the quasilinear wave equation: u tt -∆u + f (u, u t ) = 0 associated with a mixed nonhomogeneous condition[END_REF] - [START_REF] Truong | The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions[END_REF]) and references therein. In these works, many interesting results about the existence, regularity and the asymptotic behavior of solutions were obtained.

In [START_REF] Munoz -Rivera | Exponential decay of non-linear wave equation with a viscoelastic boundary condition[END_REF], J.E. Munoz-Rivera and D. Andrade dealt with the global existence and exponential decay of solutions of the nonlinear one-dimensional wave equation with a viscoelastic boundary condition.

In [START_REF] Santos | Asymptotic behavior of solutions to wave equations with a memory condition at the boundary[END_REF] - [START_REF] Santos | Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary[END_REF], Santos also studied the asymptotic behavior of solutions to a coupled system of wave equations having integral convolutions as memory terms. The main results show that solutions of that system decay uniformly in time, with rates depending on the rate of decay of the kernel of the convolutions.

In [START_REF] Truong | The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions[END_REF], the global existence and regularity of weak solutions for the linear wave equation

u tt -u xx + Ku + λu t = f (x, t), 0 < x < 1, t > 0, (1.6)
with the initial conditions as in (1.4) and the two-point boundary conditions u x (0, t) = h 0 u(0, t) + λ 0 u t (0, t) + h 1 u(1, t) + λ 1 u t (1, t) + g 0 (t),

-u x (1, t) = h 1 u(1, t) + λ 1 u t (1, t) + h 0 u(0, t) + λ 0 u t (0, t) + g 1 (t), (1.7) were proved, where h 0 , h 1 , h 0 , h 1 , λ 0 , λ 1 , λ 0 , λ 1 , K, λ are constants and u 0 , u 1 , g 0 , g 1 , f are given functions. Furthermore, the exponential decay of solutions were also given there by using Lyapunov's method.

We note more that, the following nonhomogeneous boundary conditions were considered by Hellwig ( [START_REF] Hellwig | Partial differential equations[END_REF], p.151):

   α 01 u(0, t) + α 02 u x (0, t) + α 03 u t (0, t) + β 01 u(1, t) + β 02 u x (1, t) + β 03 u t (1, t) = f 0 (t), α 11 u(0, t) + α 12 u x (0, t) + α 13 u t (0, t) + β 11 u(1, t) + β 12 u x (1, t) + β 13 u t (1, t) = f 1 (t), (1.8) where α ij , β ij , i = 0, 1, j = 1, 2, 3 are constants and f 0 (t), f 1 (t) are given functions.

Let ∆ = α 02 β 12 -α 12 β 02 = 0, (1.8) is transformed into u x (0, t) = h 0 u(0, t) + λ 0 u t (0, t) + h 1 u(1, t) + λ 1 u t (1, t) + g 0 (t),

-u x (1, t) = h 1 u(1, t) + λ 1 u t (1, t) + h 0 u(0, t) + λ 0 u t (0, t) + g 1 (t), (1.9) in which

                           h 0 = 1 ∆ (β 02 α 11 -β 12 α 01 ), h 1 = 1 ∆ (α 02 β 11 -α 12 β 01 ), λ 0 = 1 ∆ (β 02 α 13 -β 12 α 03 ), λ 1 = 1 ∆ (α 02 β 13 -α 12 β 03 ), h 0 = 1 ∆ (α 02 α 11 -α 12 α 01 ), h 1 = 1 ∆ (β 02 β 11 -β 12 β 01 ), λ 0 = 1 ∆ (α 02 α 13 -α 12 α 03 ), λ 1 = 1 ∆ (β 02 β 13 -β 12 β 03 ), g 0 (t) = 1 ∆ (β 12 f 0 (t) -β 02 f 1 (t)), g 1 (t) = 1 ∆ (α 12 f 0 (t) -α 02 f 1 (t)).
(1.10)

The main goal of this paper is to extend some results of [START_REF] Truong | The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions[END_REF]. Motivated by the problem of the exponential decay of solutions for (1.6) -(1.7), we establish a blow up result and a decay result for the general problem (1.1) - (1.4).

In Theorem 3.1, by applying techniques as in [START_REF] Messaoudi | Blow up and global existence in a nonlinear viscoelastic wave equation[END_REF] with some necessary modifications and with some restrictions on the initial data, we prove that the solution of (1.1) -(1.4) blows up in finite time.

In Theorem 4.1, by the construction of a suitable Lyapunov functional we also prove that the solution will exponential decay if the initial energy is positive and small.

The paper consists of five sections. In Section 2, we present some preliminaries and the existence results. The proofs of Theorems 3.1 and 4.1 are done in Sections 3 and 4. Finally, in Section 5 we give numerical results.

Existence and uniqueness of solution

First, we put Ω = (0, 1); Q T = Ω × (0, T ), T > 0 and we denote the usual function spaces used in this paper by the notations

C m Ω , W m,p = W m,p (Ω) , L p = W 0,p (Ω) , H m = W m,2 (Ω) , 1 ≤ p ≤ ∞, m = 0, 1, ... Let •,
• be either the scalar product in L 2 or the dual pairing of a continuous linear functional and an element of a function space. The notation || • || stands for the norm in L 2 and we denote by || • || X the norm in the Banach space X. We call X ′ the dual space of X. We denote by L p (0, T ; X), 1 ≤ p ≤ ∞ for the Banach space of the real functions u : (0, T ) → X measurable, such that

u L p (0,T ;X) = T 0 u(t) p X dt 1/p < ∞ for 1 ≤ p < ∞,
and u L ∞ (0,T ;X) = ess sup 0<t<T u(t) X for p = ∞. Let u(t), u ′ (t) = u t (t), u ′′ (t) = u tt (t), u x (t), u xx (t) denote u(x, t), ∂u ∂t (x, t), ∂ 2 u ∂t 2 (x, t), ∂u ∂x (x, t), ∂ 2 u ∂x 2 (x, t), respectively. On H 1 , we use the following norm v 1 = v 2 + v x 2 1/2
. We have the following lemmas.

Lemma 2.1. v C 0 ([0,1]) ≤ √ 2 v 1 , for all v ∈ H 1 . Lemma 2.2. Let λ 0 , λ 1 > 0 and λ 0 , λ 1 ∈ R, such that λ 0 + λ 1 < 2 √ λ 0 λ 1 . Then λ 0 x 2 + λ 1 y 2 + ( λ 0 + λ 1 )xy ≥ 1 2 µ * x 2 + y 2 , for all x, y ∈ R, (2.1)
where

µ * = 1 4 -( λ 0 + λ 1 ) 2 + 4λ 0 λ 1 min 1 λ 0 , 1 λ 1 > 0. (2.
2)

The proofs of these lemmas are straightforward. We shall omit the details.

Next, we state two local existence theorems. We make the following assumptions: Suppose that p, α, β, λ, λ 0 , λ 1 , λ 0 , λ 1 ∈ R, are constants satisfying

(A 1 ) p > 2, α > 2, β > 2, λ > 0; (A 2 ) λ 0 , λ 1 > 0, λ 0 , λ 1 ∈ R, with λ 0 + λ 1 < 2 √ λ 0 λ 1 . Let (A 3 ) h i ∈ H 1 (0, T ) , i = 1, 2.
Then we have the following theorem about the existence of a "strong solution". Theorem 2.3. Suppose that (A 1 ) -(A 3 ) hold and the initial data (u 0 , u 1 ) ∈ H 2 × H 1 satisfies the compatibility conditions

u 0x (0) = -|u 0 (0)| α-2 u 0 (0) + λ 0 u 1 (0) + h 1 (0)u 0 (1) + λ 1 u 1 (1), -u 0x (1) = -|u 0 (1)| β-2 u 0 (1) + λ 1 u 1 (1) + h 0 (0)u 0 (0) + λ 0 u 1 (0). (2.3) Then problem (1.1) -(1.4) has a unique local solution u ∈ L ∞ 0, T * ; H 2 , u t ∈ L ∞ 0, T * ; H 1 , u tt ∈ L ∞ 0, T * ; L 2 , u(0, •), u(1, •) ∈ H 2 (0, T * ) , (2.4)
for T * > 0 small enough.

Remark 2.1.

The regularity obtained by (2.4) shows that problem (1.1) -(1.4) has a unique strong solution

             u ∈ L ∞ 0, T * ; H 2 ∩ C 0 0, T * ; H 1 ∩ C 1 0, T * ; L 2 , u t ∈ L ∞ 0, T * ; H 1 ∩ C 0 0, T * ; L 2 , u tt ∈ L ∞ 0, T * ; L 2 , u(i, •) ∈ H 2 (0, T * ) , i = 0, 1.
(2.5)

With less regular initial data, we obtain the following theorem about the existence of a weak solution.

Theorem 2.4. Suppose that

(A 1 ) -(A 3 ) hold. Let (u 0 , u 1 ) ∈ H 1 × L 2 . Then problem (1.1) -(1.4) has a unique local solution u ∈ C [0, T * ]; H 1 ∩ C 1 [0, T * ]; L 2 , u(i, •) ∈ H 1 (0, T * ) , i = 0, 1, (2.6)
for T * > 0 small enough.

Proof of Theorem 2.3.

The proof is established by a combination of the arguments in [START_REF] Truong | The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions[END_REF]. It consits of steps 1 -4.

Step 1. The Faedo-Galerkin approximation. Let {w j } be a denumerable base of H 1 . We find the approximate solution of the problem (1.1) - (1.4) in the form

u m (t) = m j=1 c mj (t)w j , (2.7)
where the coefficient functions c mj , 1 ≤ j ≤ m, satisfy the system of ordinary differential equations

                             u ′′ m (t), w j + u mx (t), w jx + u m (t), w j + λ u ′ m (t), w j + λ 0 u ′ m (0, t) + h 1 (t)u m (1, t) + λ 1 u ′ m (1, t) w j (0) + λ 1 u ′ m (1, t) + h 0 (t)u m (0, t) + λ 0 u ′ m (0, t) w j (1) = |u m | p-2 u m , w j + |u m (0, t)| α-2 u m (0, t)w j (0) + |u m (1, t)| β-2 u m (1, t)w j (1), 1 ≤ j ≤ m, u m (0) = u 0 , u ′ m (0) = u 1 .
(2.8)

From the assumptions of Theorem 2.3, system (2.8) has a solution u m on an interval [0,

T m ] ⊂ [0, T ].
Step 2. The first estimate. Multiplying the j th equation of (2.8) by c ′ mj (t) and summing up with respect to j, afterwards, integrating by parts with respect to the time variable from 0 to t, after some rearrangements and using Lemma 2.2, we get

S m (t) ≤ S m (0) + 2 t 0 |u m (s)| p-2 u m (s), u ′ m (s) ds +2 t 0 |u m (0, s)| α-2 u m (0, s)u ′ m (0, s)ds + 2 t 0 |u m (1, s)| β-2 u m (1, s)u ′ m (1, s)ds -2 t 0 h 1 (s)u m (1, s)u ′ m (0, s)ds -2 t 0 h 0 (s)u m (0, s)u ′ m (1, s)ds, (2.9)
where

S m (t) = u ′ m (t) 2 + u m (t) 2 1 + 2λ t 0 u ′ m (s) 2 ds + µ * t 0 u ′ m (0, s) 2 + u ′ m (1, s) 2 ds, (2.10) S m (0) = u 1 2 + u 0 2 1 ≡ S 0 . (2.11)
Applying the classical inequalities, we estimate the terms on the right-hand side of (2.9) and obtain

S m (t) ≤ d 0 + d 1 t 0 (S m (s)) p 2 ds + d 2 t 0 (S m (s)) α-1 ds +d 3 t 0 (S m (s)) β-1 ds + d 4 (T ) t 0 S m (s)ds, 0 ≤ t ≤ T m , (2.12) where            d 0 = 2S 0 , d 1 = 4 √ 2 p-1 , d 2 = 1 µ * 2 α+3 , d 3 = 1 µ * 2 β+3 , d 4 (T ) = 32 µ * h 0 2 L ∞ (0,T ) + h 1 2 L ∞ (0,T ) , p 2 > 1, α -1 > 1, β -1 > 1.
(2.13)

Then, by solving a nonlinear Volterra integral equation (based on the methods in [START_REF] Lakshmikantham | Differential and Integral Inequalities[END_REF]), we get the following lemma.

Lemma 2.5. There exists a constant T * > 0 depending on T (independent of m) such that

S m (t) ≤ C T , ∀m ∈ N, ∀t ∈ [0, T * ], (2.14)
where C T is a constant depending only on T.

Lemma 2.5 allows one to take constant T m = T * for all m.

The second estimate.

First of all, we estimate u ′′ m (0). By taking t = 0 and w j = u ′′ m (0) in (??), we assert

u ′′ m (0) ≤ u 0xx + u 0 + λ u 1 + |u 0 | p-1 = X * 0 . (2.15)
Now, by differentiating (2.8) with respect to t and substituting w j = u ′′ m (t), after integrating with respect to the time variable from 0 to t, using again Lemma 2.2, we have

X m (t) ≤ X m (0) -2 t 0 h 1 (s)u ′ m (1, s) + h ′ 1 (s)u m (1, s) u ′′ m (0, s)ds -2 t 0 h 0 (s)u ′ m (0, s) + h ′ 0 (s)u m (0, s) u ′′ m (1, s)ds +2(α -1) t 0 |u m (0, s)| α-2 u ′ m (0, s)u ′′ m (0, s)ds +2(β -1) t 0 |u m (1, s)| β-2 u ′ m (1, s)u ′′ m (1, s)ds +2(p -1) t 0 |u m (s)| p-2 u ′ m (s), u ′′ m (s) ds, (2.16)
where

X m (t) = u ′′ m (t) 2 + u ′ m (t) 2 1 + 2λ t 0 u ′′ m (s) 2 ds + µ * t 0 u ′′ m (0, s) 2 + u ′′ m (1, s) 2 ds, (2.17) X m (0) = u ′′ m (0) 2 + u 1 2 1 ≤ X * 0 2 + u 1 2 1 ≡ X 0 . (2.18)
Estimate respectively all the terms on the right-hand side of (2.16) leads to

X m (t) ≤ d T + 2 t 0 X m (s)ds, (2.19)
where

d T = 2X 0 + 16 µ * (α -1) 2 2 α-2 C α-1 T + (β -1) 2 2 β-2 C β-1 T +(p -1) 2 2 p-1 T C p-1 T + 32C T µ * d T h 0 2 H 1 (0,T ) + h 1 2 H 1 (0,T ) , (2.20) in which d T is a constant verifying the inequality 1 µ * v 2 L ∞ (0,T ) + 2 v ′ 2 L 2 (0,T ) ≤ d T v 2 H 1 (0,T ) , for all v ∈ H 1 (0, T ).
By Gronwall's lemma, it follows from (2.19), that

X m (t) ≤ d T exp(2T ) ≤ C T , ∀t ∈ [0, T * ], (2.21)
where C T is a constant depending only on T.

Step 3. Limiting process. From (2.10), (2.14), (2.17) and (2.21), we deduce the existence of a subsequence of {u m } still also so denoted, such that

                   u m → u in L ∞ (0, T * ; H 1 ) weakly*, u ′ m → u ′ in L ∞ (0, T * ; H 1 ) weakly*, u ′′ m → u ′′ in L ∞ (0, T * ; L 2 ) weakly*, u m (0, •) → u(0, •) in H 2 (0, T * ) weakly, u m (1, •) → u(1, •) in H 2 (0, T * )
weakly.

(2.22)

By the compactness lemma of Lions ([5], p. 57) and the compact imbedding

H 2 (0, T * ) ֒→ C 1 ([0, T * ]) , we can deduce from (2.22) the existence of a subsequence still denoted by {u m }, such that        u m → u strongly in L 2 (Q T * ) and a.e. in Q T * , u ′ m → u ′ strongly in L 2 (Q T * ) and a.e. in Q T * , u m (i, •) → u(i, •) strongly in C 1 ([0, T * ]) , i = 0, 1.
(2.23)

Using the following inequality

|x| p-2 x -|y| p-2 y ≤ (p -1)M p-2 |x -y| , ∀x, y ∈ [-M, M ], ∀M > 0, ∀p ≥ 2, (2.24) with M = √ 2C T , we deduce from (2.14) that |u m | p-2 u m -|u| p-2 u ≤ (p -1)M p-2 |u m -u| , for all m, (x, t) ∈ Q T * . (2.25)
Hence, by (2.23) 1 , we deduce from (2.25), that

|u m | p-2 u m → |u| p-2 u strongly in L 2 (Q T * ). ( 2
                       u ′′ (t), v + u x (t), v x + u(t), v + λ u ′ (t), v + λ 0 u ′ (0, t) + h 1 (t)u(1, t) + λ 1 u ′ (1, t) v(0) + λ 1 u ′ (1, t) + h 0 (t)u(0, t) + λ 0 u ′ (0, t) v(1) = |u| p-2 u, v + |u(0, t)| α-2 u(0, t)v(0) + |u(1, t)| β-2 u(1, t)v(1), for all v ∈ H 1 , u(0) = u 0 , u ′ (0) = u 1 .
(2.27)

On the other hand, we have from (2.22) 1,2,3 , (2.27) 1 that

u xx = u ′′ + u + λu ′ -|u| p-2 u ∈ L ∞ (0, T * ; L 2 ).
(2.28) Thus u ∈ L ∞ (0, T * ; H 2 ) and the existence of the solution is proved completely.

Step 4. Uniqueness of the solution. Let u 1 , u 2 be two weak solutions of problem (1.1) -(1.4), such that

u i ∈ L ∞ 0, T * ; H 2 , u ′ i ∈ L ∞ 0, T * ; L 2 , u ′′ i ∈ L ∞ 0, T * ; L 2 , u i (0, •), u i (1, •) ∈ H 2 (0, T * ) , i = 1, 2.
(2.29)

Then w = u 1 -u 2 verifies                                      w ′′ (t), v + w x (t), v x + w(t), v + λ w ′ (t), v + λ 0 w ′ (0, t) + h 1 (t)w(1, t) + λ 1 w ′ (1, t) v(0) + λ 1 w ′ (1, t) + h 0 (t)w(0, t) + λ 0 w ′ (0, t) v(1) = |u 1 | p-2 u 1 -|u 2 | p-2 u 2 , v + |u 1 (0, t)| α-2 u 1 (0, t) -|u 2 (0, t)| α-2 u 2 (0, t) v(0) + |u 1 (1, t)| β-2 u 1 (1, t) -|u 2 (1, t)| β-2 u 2 (1, t) v(1), for all v ∈ H 1 , w(0) = w ′ (0) = 0.
(2.30)

We take v = w = u 1 -u 2 in (2.30) and integrating with respect to t, we obtain

S(t) ≤ -2 t 0 h 0 (s)w(0, s)w ′ (1, s)ds -2 t 0 h 1 (s)w(1, s)w ′ (0, s)ds +2 t 0 |u 1 (0, s)| α-2 u 1 (0, s) -|u 2 (0, s)| α-2 u 2 (0, s) w ′ (0, s)ds +2 t 0 |u 1 (1, s)| β-2 u 1 (1, s) -|u 2 (1, s)| β-2 u 2 (1, s) w ′ (1, s)ds +2 t 0 |u 1 (s)| p-2 u 1 (s) -|u 2 (s)| p-2 u 2 (s), w ′ (s) ds, (2.31)
where

S(t) = w ′ (t) 2 + w(t) 2 1 + 2λ t 0 w ′ (s) 2 ds + µ * t 0 |w ′ (0, s)| 2 + |w ′ (1, s)| 2 ds. (2.32) It implies that S(t) ≤ K M t 0 S(s)ds, (2.33)
where

K M = 32 µ * h 0 2 L ∞ (0,T ) + h 1 2 L ∞ (0,T ) + (α -1) 2 M 2α-4 1 + (β -1) 2 M 2β-4 1 + 2(p -1)M p-2 1 , (2.34) with M 1 = √ 2 u L ∞ (0,T * ;H 1 ) + v L ∞ (0,T * ;H 1 )
. By Gronwall's lemma, it follows from (2.23), that S ≡ 0, i.e., u ≡ v. Theorem 2.3 is proved completely.

Proof of Theorem 2.4.

In order to obtain the existence of a weak solution, we use standard arguments of density.

Let us consider (u 0 , u 1 ) ∈ H 1 × L 2 and let sequences {u 0m } and {u 1m } in H 2 and H 1 , respectively, such that

u 0m → u 0 strongly in H 1 , u 1m → u 1 strongly in L 2 . (2.35)
So {(u 0m , u 1m )} satisfy, for all m ∈ N, the compatibility conditions

u 0mx (0) = -|u 0m (0)| α-2 u 0m (0) + λ 0 u 1m (0) + h 1 (0)u 0m (1) + λ 1 u 1m (1), -u 0mx (1) = -|u 0m (1)| β-2 u 0m (1) + λ 1 u 1m (1) + h 0 (0)u 0m (0) + λ 0 u 1m (0).
(2.36)

Then, for each m ∈ N there exists a unique function u m in the conditions of the Theorem 2.3. So we can verify

                             u ′′ m (t), v + u mx (t), v x + u m (t), v + λ u ′ m (t), v + λ 0 u ′ m (0, t) + h 1 (t)u m (1, t) + λ 1 u ′ m (1, t) v(0) + λ 1 u ′ m (1, t) + h 0 (t)u m (0, t) + λ 0 u ′ m (0, t) v(1) = |u m | p-2 u m , v + |u m (0, t)| α-2 u m (0, t)v(0) + |u m (1, t)| β-2 u m (1, t)v(1), for all v ∈ H 1 , u m (0) = u 0m , u ′ m (0) = u 1m , (2.37)
and

             u m ∈ L ∞ 0, T * ; H 2 ∩ C 0 0, T * ; H 1 ∩ C 1 0, T * ; L 2 , u ′ m ∈ L ∞ 0, T * ; H 1 ∩ C 0 0, T * ; L 2 , u ′′ m ∈ L ∞ 0, T * ; L 2 , u m (0, •), u m (1, •) ∈ H 2 (0, T * ) .
(2.38) By the same arguments used to obtain the above estimates, we get

u ′ m (t) 2 + u m (t) 2 1 + 2λ t 0 u ′ m (s) 2 ds + µ * t 0 |u ′ m (0, s)| 2 + |u ′ m (1, s)| 2 ds ≤ C T , (2.39)
∀t ∈ [0, T * ], where C T is a positive constant independent of m and t.

On the other hand, we put w m,l = u m -u l , from (2.37), it follows that

                                 w ′′ m,l (t), v + w m,lx (t), v x + w m,l (t), v + λ w ′ m,l (t), v + λ 0 w ′ m,l (0, t) + h 1 (t)w m,l (1, t) + λ 1 w ′ m,l (1, t) v(0) + λ 1 w ′ m,l (1, t) + h 0 (t)w m,l (0, t) + λ 0 w ′ m,l (0, t) v(1) = |u m | p-2 u m -|u l | p-2 u l , v + |u m (0, t)| α-2 u m (0, t) -|u l (0, t)| α-2 u l (0, t) v(0) + |u m (1, t)| β-2 u m (1, t) -|u l (1, t)| β-2 u l (1, t) v(1), for all v ∈ H 1 , w m,l (0) = u 0m -u 0l , w ′ m,l (0) = u 1m -u 1l .
(2.40)

We take v = w ′ m,l = u ′ m -u ′ l , in (2.40
) and integrating with respect to t, we obtain

S m,l (t) ≤ S m,l (0) -2 t 0 h 1 (s)w m,l (1, s)w ′ m,l (0, s)ds -2 t 0 h 0 (s)w m,l (0, s)w ′ m,l (1, s)ds +2 t 0 |u m (0, s)| α-2 u m (0, s) -|u l (0, s)| α-2 u l (0, s) w ′ m,l (0, s)ds +2 t 0 |u m (1, s)| β-2 u m (1, s) -|u l (1, s)| β-2 u l (1, s) w ′ m,l (1, s)ds +2 t 0 |u m (s)| p-2 u m (s) -|u l (s)| p-2 u l (s), w ′ m,l (s) ds, (2.41)
where

S m,l (t) = w ′ m,l (t) 2 + w m,l (t) 2 1 + 2λ t 0 w ′ m,l (s) 2 ds + µ * t 0 w ′ m,l (0, s) 2 + w ′ m,l (1, s) 2 ds, (2.42) S m,l (0) = u 1m -u 1l 2 + u 0m -u 0l 2 1 . (2.43) Hence S m,l (t) ≤ 2 u 1m -u 1l 2 + u 0m -u 0l 2 1 + K T t 0 S m,l (s)ds, (2.44)
where

K T = 2(p -1)M p-2 T + 32 µ * h 1 2 L ∞ (0,T ) + h 0 2 L ∞ (0,T ) + (α -1) 2 M 2α-4 T + (β -1) 2 M 2β-4 T , (2.45) with M T = √ 2C T . By Gronwall's lemma, it follows from (2.44), that S m,l (t) ≤ 2 u 1m -u 1l 2 + u 0m -u 0l 2 1 exp(T K T ), ∀t ∈ [0, T * ] (2.46)
Convergences of the sequences {u 0m }, {u 1m } imply the convergence to zero (when m, l → ∞) of terms on the right hand side of (2.46). Therefore, we get

u m → u strongly in C 0 ([0, T * ]; H 1 ) ∩ C 1 ([0, T * ]; L 2 ), u m (i, •) → u(i, •) strongly in H 1 (0, T * ), i = 0, 1. (2.47)
On the other hand, from (2.39), we deduce the existence of a subsequence of {u m } still also so denoted, such that

                         u m → u in L ∞ (0, T * ; H 1 ) weakly*, u ′ m → u ′ in L ∞ (0, T * ; L 2 ) weakly*, u m (0, •) → u(0, •) in H 1 (0, T * ) weakly, u m (1, •) → u(1, •) in H 1 (0, T * ) weakly, u ′ m (0, •) → u ′ (0, •) in L 2 (0, T * ) weakly, u ′ m (1, •) → u ′ (1, •) in L 2 (0, T * ) weakly.
(2.48)

By the compactness lemma of Lions ([5], p. 57) and the compact imbedding H 1 (0, T * ) ֒→ C 0 ([0, T * ]) , we can deduce from (2.48) 1-4 the existence of a subsequence still denoted by {u m }, such that

u m → u strongly in L 2 (Q T * ) and a.e. in Q T * , u m (i, •) → u(i, •) strongly in C 0 ([0, T * ]) , i = 0, 1. ( 2
                      d dt u ′ (t), v + u x (t), v x + u(t), v + λ u ′ (t), v + λ 0 u ′ (0, t) + h 1 (t)u(1, t) + λ 1 u ′ (1, t) v(0) + λ 1 u ′ (1, t) + h 0 (t)u(0, t) + λ 0 u ′ (0, t) v(1) = |u| p-2 u, v + |u(0, t)| α-2 u(0, t)v(0) + |u(1, t)| β-2 u(1, t)v(1), for all v ∈ H 1 , u(0) = u 0 , u ′ (0) = u 1 .
(2.51)

Next, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions. See for example Ngoc et al. [START_REF] Ngoc | On a nonlinear wave equation associated with the boundary conditions involving convolution[END_REF].

Theorem 2.4 is proved completely.

Remark 2.2. In case 1 < p, α, β ≤ 2, and h 0 , h 1 ∈ L ∞ (0, T ) , (u 0 , u 1 ) ∈ H 1 × L 2 , the integral inequality (??) leads to the following global estimation

S m (t) ≤ C T , ∀m ∈ N, ∀t ∈ [0, T ], ∀T > 0.
(2.52)

Then, by applying a similar argument used in the proof of Theorem 2.4, we can obtain a global weak solution

u of problem (1.1) -(1.4) satisfying u ∈ L ∞ 0, T ; H 1 , u t ∈ L ∞ 0, T ; L 2 , u(i, •) ∈ H 1 (0, T ) , i = 0, 1.
(2.53) However, in case 1 < p, α, β < 2, we do not imply that a weak solution obtained here belongs to C [0, T ]; H 1 ∩ C 1 [0, T ]; L 2 . Furthermore, the uniqueness of a weak solution is also not asserted.

Finite time blow up

In this section we show that the solution of problem (1.1) -(1.4) blows up in finite time if

λ 0 = λ 1 = λ, with λ < √ λ 0 λ 1 , and
-H(0) = 1 2 u 1 2 + 1 2 u 0 2 1 -1 p u 0 p L p -1 α |u 0 (0)| α -1 β |u 0 (1)| β + hu 0 (0)u 0 (1) < 0. (3.1)
First, in order to obtain the blow up result, we make the following assumptions

(A ′ 2 ) λ 0 = λ 1 = λ, with λ < √ λ 0 λ 1 . (A ′ 3 ) h 0 (t) = h 1 (t) = h
, where h is a constant satisfies h < q-2 4(q+2) , q = min{p, α, β};

Then we obtain the theorem.

Theorem 3.1. Let the assumptions (A 1 ), (A ′ 2 ), (A ′ 3 ) hold and H(0) > 0. Then, for any (u 0 , u 1 ) ∈ H 1 × L 2 , the solution u of problem (1.1) -(1.4) blows up in finite time.

Proof. We denote by E(t) the energy associated to the solution u, defined by

E(t) = 1 2 u ′ (t) 2 + 1 2 u(t) 2 1 - 1 p u(t) p L p - 1 α |u(0, t)| α - 1 β |u(1, t)| β , (3.2)
and we put

H(t) = -E(t) -hu(0, t)u(1, t). (3.3)
From Lemma 2.1, it is easy to see that

H(t) ≥ 1 p u(t) p L p + 1 α |u(0, t)| α + 1 β |u(1, t)| β - 1 2 u ′ (t) 2 - 1 2 + 2 h u(t) 2 1 . (3.4)
On the other hand, by multiplying (1.1) by u ′ (x, t) and integrating over [0, 1], we get

H ′ (t) = λ u ′ (t) 2 + λ 0 u ′ (0, t) 2 + λ 1 u ′ (1, t) 2 + 2 λu ′ (0, t)u ′ (1, t) ≥ 0, ∀t ∈ [0, T * ). (3.5)
By Lemma 2.2, we have

λ 0 u ′ (0, t) 2 + λ 1 u ′ (1, t) 2 + 2 λu ′ (0, t)u ′ (1, t) ≥ 1 2 µ * u ′ (0, t) 2 + u ′ (1, t) 2 , ∀t ∈ [0, T * ), (3.6)
where

µ * = λ 0 λ 1 -λ 2 min 1 λ 0 , 1 λ 1 > 0. (3.7)
Hence, we can deduce from (3.5), (3.6) and

H(0) > 0 that 0 < H(0) ≤ H(t) ≤ 1 p u(t) p L p + 1 α |u(0, t)| α + 1 β |u(1, t)| β , ∀t ∈ [0, T * ). (3.8)
Now, we define the functional

L(t) = H 1-η (t) + εΦ(t), (3.9)
where

Φ(t) = u(t), u ′ (t) + λ 2 u(t) 2 + λ 0 2 |u(0, t)| 2 + λ 1 2 |u(1, t)| 2 + λu(0, t)u(1, t), (3.10)
for ε small enough and

0 < η ≤ p -2 2p < 1 2 . ( 3
Φ ′ (t) = u ′ (t) 2 + u(t) p L p + |u(0, t)| α + |u(1, t)| β -u(t) 2 1 -2 hu(0, t)u(1, t). (3.13)
By taking a derivative of (3.9) and using (3.13), we obtain

L ′ (t) = (1 -η)H -η (t)H ′ (t) + ε u ′ (t) 2 + ε u(t) p L p + ε |u(0, t)| α + |u(1, t)| β -ε u(t) 2 1 -2ε hu(0, t)u(1, t).
-2 hu(0, t)u(1, t) ≥ -4 h u(t) 2 1 , (3.15)
we deduce that

L ′ (t) ≥ [λ(1 -η)H -η (t) + ε] u ′ (t) 2 + ε u(t) p L p + |u(0, t)| α + |u(1, t)| β -ε 1 + 4 h u(t) 2 1 . (3.16)
On the other hand, it follows from (3.8) and the following inequality

H(t) ≤ 1 p u(t) p L p + 1 α |u(0, t)| α + 1 β |u(1, t)| β - 1 2 u ′ (t) 2 - 1 2 -2 h u(t) 2 1 , (3.17)
that u(t) 2 1 ≤ 2 q 1 1 -4 h u(t) p L p + |u(0, t)| α + |u(1, t)| β , (3.18)
where q = min{p, α, β}. Combining (3.16) and (3.18), we have

L ′ (t) ≥ ε u ′ (t) 2 + ε 1 -2 q 1+4| h| 1-4| h| u(t) p L p + |u(0, t)| α + |u(1, t)| β . (3.19)
Using the inequality

u(t) p L p + |u(0, t)| α + |u(1, t)| β ≥ qH(t), t ≥ 0, (3.20)
we can deduce from (3.19) that, with ε is small enough,

L ′ (t) ≥ d 1 H(t) + u ′ (t) 2 + u(t) p L p + |u(0, t)| α + |u(1, t)| β , (3.21)
for d 1 is a positive constant. The lemma 3.2 is proved completely.

Remark 3.1. From the formula of L(t) and the Lemma 3.2, we can choose ε small enough such that

L(t) ≥ L(0) > 0, ∀t ∈ [0, T * ). (3.22)
Now we continue to prove Theorem 3.1.

Using the inequality 6 i=1 x i p ≤ 6 p-1 6 i=1 x p i , for all p > 1, and x 1 , ..., x 6 ≥ 0, (

we deduce from (3.9), (3.10) that

L 1/(1-η) (t) ≤ Const H(t) + | u(t), u ′ (t) | 1/(1-η) + u(t) 2/(1-η) +|u(0, t)| 2/(1-η) + |u(1, t)| 2/(1-η) + |u(0, t)u(1, t)| 1/(1-η) ≤ Const H(t) + | u(t), u ′ (t) | 1/(1-η) + |u(0, t)| 2/(1-η) + |u(1, t)| 2/(1-η) + u(t) 2/(1-η) L p . (3.24)
On the other hand, by using the Young's inequality

| u(t), u ′ (t) | 1/(1-η) ≤ u(t) 1/(1-η) u ′ (t) 1/(1-η) ≤ Const u(t) 1/(1-η) L p u ′ (t) 1/(1-η) ≤ Const u(t) s L p + u ′ (t) 2 , (3.25)
where s = 2/(1 -2η) ≤ p by (3.11). Now, we need the following lemma.

Lemma 3.3. Let 2 ≤ r 1 ≤ p, 2 ≤ r 2 ≤ α, 2 ≤ r 3 ≤ β, we have v r 1 L p + |v(0)| r 2 + |v(1)| r 3 ≤ 5 v 2 1 + v p L p + |v(0)| α + |v(1)| β , (3.26)
for any v ∈ H 1 .

Proof of Lemma 3.3.

(i) We consider two cases for v L p : (i.1) Case 1:

v L p ≤ 1 : By 2 ≤ r 1 ≤ p, we have v r 1 L p ≤ v 2 L p ≤ v 2 1 ≤ v 2 1 + v p L p + |v(0)| α + |v(1)| β ≡ ρ[v]. (3.27)
(i.2) Case 2: v L p ≥ 1 : By 2 ≤ r 1 ≤ p, we have v r 1 L p ≤ v p L p ≤ ρ[v]. (3.28) Therefore v r 1 L p ≤ v p L p ≤ ρ[v], for any v ∈ H 1 . (3.29) (ii) We consider two cases for |v(0)| : (ii.1) Case 1: |v(0)| ≤ 1 : By 2 ≤ r 2 ≤ α, we have |v(0)| r 1 ≤ |v(0)| 2 ≤ v 2 C 0 ([0,1]) ≤ 2 v 2 1 ≤ 2ρ[v]. (3.30) (ii.2) Case 2: |v(0)| ≥ 1 : By 2 ≤ r 2 ≤ α, we have |v(0)| r 1 ≤ |v(0)| α ≤ ρ[v]. (3.31)
Therefore

|v(0)| r 1 ≤ 2ρ[v], for any v ∈ H 1 . (3.32)
(iii) Similarly |v(1)| r 2 ≤ 2ρ[v], for any v ∈ H 1 . (3.33)
Combining (3.29), (3.32), (3.33), we obtain

v r 1 L p + |v(0)| r 2 + |v(1)| r 3 ≤ 5ρ[v] ≤ 5 v 2 1 + v p L p + |v(0)| α + |v(1)| β , ∀v ∈ H 1 . ( 3
L 1/(1-η) (t) ≤ Const H(t) + u ′ (t) 2 + u(t) p L p + |u(0, t)| α + |u(1, t)| β , ∀t ∈ [0, T * ). (3.35) This implies that L ′ (t) ≥ d 2 L 1/(1-η) (t), ∀t ∈ [0, T * ), (3.36)
where d 2 is a positive constant. By integrating (3.36) over (0, t) we deduce that

L η/(1-η) (t) ≥ 1 L -η/(1-η) (0) -d 2 η 1-η t , 0 ≤ t < 1 -η d 2 η L -η/(1-η) (0). (3.37)
Therefore, (3.37) shows that L(t) blows up in a finite time given by

T * = 1 -η d 2 η L -η/(1-η) (0). ( 3

Exponential decay

In this section we show that each solution u of (1.1) -(1.4) is global and exponential decay provided that I(0) = u 0 2 1 -u 0 p L p -|u 0 (0)| α -|u 0 (1)| β > 0 and E(0) is small enough. First, we construct the following Lyapunov functional

L(t) = E(t) + δψ(t), (4.1)
where δ > 0 is chosen later and

ψ(t) = u(t), u ′ (t) + λ 2 u(t) 2 + λ 0 2 |u(0, t)| 2 + λ 1 2 |u(1, t)| 2 . (4.2) Put I(t) = I(u(t)) = u(t) 2 1 -u(t) p L p -|u(0, t)| α -|u(1, t)| β . (4.3)
We make the following assumption

(A ′′ 3 ) h i ∈ L ∞ (R + ) ∩ L 2 (R + ) , i = 1, 2.
Then we have the following theorem.

Theorem 4.1. Assume that (A 1 ), (A 2 ), (A ′′ 3 ) hold. Let I(0) > 0 and the initial energy E(0) satisfies

η * = C p p 2qr q-2 E(0) (p-2)/2 + 2 α/2 2qr q-2 E(0) (α-2)/2 + 2 β/2 2qr q-2 E(0) (β-2)/2 < 1, (4.4)
with q = min{p, α, β}, and

r = exp 4q µ * (q -2) h 0 2 L 2 (R + ) + h 1 2 L 2 (R + ) , (4.5)
and C p is a constant verifying the inequality v L p ≤ C p v 1 , for all v ∈ H 1 .

Then, there exist positive constants C, γ such that, for

h 0 L 2 (R + ) , h 1 L 2 (R + )
sufficiently small, we have E(t) ≤ C exp(-γt), for all t ≥ 0. (4.6)

Proof.

First, we need the following lemmas Lemma 4.2. The energy functional E(t) satisfies

E ′ (t) ≤ -λ u ′ (t) 2 + 1 2µ * h 2 0 (t) + h 2 1 (t) u (t) 2 1 -1 4 µ * |u ′ (0, t)| 2 + |u ′ (1, t)| 2 . (4.7)
Proof of Lemma 4.2. Multiplying (1.1) by u ′ (x, t) and integrating over [0, 1], we get

E ′ (t) = -λ u ′ (t) 2 -λ 0 |u ′ (0, t)| 2 + λ 1 |u ′ (1, t)| 2 + λ 0 + λ 1 u ′ (0, t) u ′ (1, t) -h 0 (t)u (0, t) u ′ (1, t) -h 1 (t)u (1, t) u ′ (0, t) . (4.8)
Again, by lemma 2.2, we have

λ 0 |u ′ (0, t)| 2 + λ 1 |u ′ (1, t)| 2 + λ 0 + λ 1 u ′ (0, t) u ′ (1, t) ≥ 1 2 µ * |u ′ (0, t)| 2 + |u ′ (1, t)| 2 .
(4.9)

where

β 2 = 1 + δ + 2q q -2 1 2 + δ 1 + λ 2 + λ 0 + λ 1 . (4.21)
Similarly, we can prove that

L(t) ≥ 1-δ 2 u ′ (t) 2 + 1 2 (1 -δ) u(t) 2 1 -1 p u(t) p L p -1 α |u(0, t)| α -1 β |u(1, t)| β ≥ 1-δ 2 u ′ (t) 2 + 1 2 q-2 q -δ u(t) 2 1 + 1 q I(t) ≥ β 1 E(t), (4.22)
where

β 1 = min 1 -δ; q -2 q -δ > 0, δ is small enough. ( 4
′ (t) ≤ u ′ (t) 2 -1 -η * -ε 1 -2 h 0 (t) + h 1 (t) u (t) 2 1 + 1 ε 1 λ 2 0 + λ 2 1 |u ′ (0, t)| 2 + |u ′ (1, t)| 2 . (4.24)
for all ε 1 > 0.

Proof of Lemma 4.5. By multiplying (1.1) by u(x, t) and integrating over [0, 1], we obtain

ψ ′ (t) = u ′ (t) 2 + u(t) p L p + |u(0, t)| α + |u(1, t)| β -u (t) 2 1 -h 0 (t) + h 1 (t) u (0, t) u (1, t) -λ 0 u ′ (0, t) u (1, t) -λ 1 u (0, t) u ′ (1, t) . (4.25)
Hence, the lemma 4.5 is proved by using some simple estimates. Now we continue to prove Theorem 4.1.

It follows from (4.1), (4.2), (4.7) and (4.24), that

L ′ (t) ≤ -(λ -δ) u ′ (t) 2 + 2 µ * h 2 0 (t) + h 2 1 (t) + 2δ h 0 (t) + h 1 (t) -δ(1 -η * -ε 1 ) u (t) 2 1 -1 4 µ * -δ ε 1 λ 2 0 + λ 2 1 |u ′ (0, t)| 2 + |u ′ (1, t)| 2 (4.26) for all δ, ε 1 > 0. Let 0 < ε 1 < 1 -η * . (4.27)
Then, for δ small enough, with 0 < δ < λ and if h 0 , h 1 satisfy

2 µ * h 0 2 L ∞ (R + ) + h 1 2 L ∞ (R + ) + 2δ h 0 L ∞ (R + ) + h 1 L ∞ (R + ) < δ(1 -η * -ε 1 ), ( 4
                                                                dU (m) j dt (t) = V (m) j (t), j = 0, 1, ..., N, dV (m) 0 dt (t) = -1 + N 2 U (m) 0 (t) + N 2 U (m) 1 (t) -N h 1 (t)U (m) N (t) -(λ + N λ 0 )V (m) 0 (t) -N λ 1 V (m) N (t) + U (m-1) 0 p-2 U (m-1) 0 + N U (m-1) 0 α-2 U (m-1) 0 -N g 0 (t) + f 0 (t), dV (m) j dt (t) = N 2 U (m) j-1 (t) -1 + 2N 2 U (m) j (t) + N 2 U (m) j (t) -λV (m) j (t) + U (m-1) j p-2 U (m-1) j + f j (t), j = 1, N -1, dV (m) N dt (t) = -N h 0 (t)U (m) 0 (t) + N 2 U (m) N -1 (t) -1 + N 2 U (m) N (t) -N λ 0 V (m) 0 (t) -(λ + N λ 1 )V (m) N (t) + U (m-1) N p-2 U (m-1) N + N U (m-1) N β-2 U (m-1) N -N g 1 (t) + f N (t), U (m) j (0) = u 0 (x j ), V (m) j (0) = u 1 (x j ), j = 0, N , m = 1, 2, .... (5.7)
Then system (5.7) is equivalent to: -N g 1 (t) + f N (t), f j (t) = f (x j , t), j = 0, N .

d dt                    U (m) 0 U (m) 1 . . . U (m) N V (m) 0 V (m) 1 . . . . . . V (m) N                    =                  0 0 • • • • • • 0 1 0 0 • • • • • • 0 1 . . . . . . • • • • • • . . . . . . 0 0 • • • • • • 0 1 γ + α 1 α 1 γ 1 (t) δ 0 δ 1 α 1 γ α 1 -λ . . . . . . . . . . . . α 1 γ α 1 -λ γ 0 (t) α 1 γ + α 1 δ 0 δ 1                                     U (m) 0 U (m) 1 . . .
(5.9)

Rewritten (5.8) d dt X (m) (t) = A(t)X (m) (t) + F (m) (t, X (m-1) ),

X (m) (0) = X 0 , (5.10)
where

                       X (m) (t) = U

(3. 14 )

 14 Since (3.5), (3.14) and the following inequality

  .34) Lemma 3.3 is proved completely. Combining (3.18), (3.24) -(3.26) and using the Lemma 3.2 we obtain

  .38) Theorem 3.1 is proved completely.

U

FNF+F-F

  (0) = ( u 0 (x 0 ), u 0 (x 1 ), , ..., u 0 (x N )) , ..., V (m)N (0) = ( u 1 (x 0 ), u 1 (x 1 ), , ..., u 1 (x N )) , where                                          α 1 = N 2 , γ = -1 -2N 2 = -1 -2α 1 , γ 0 (t) = -N h 0 (t), γ 1 (t) = -N h 1 (t), δ 0 = -λ -N λ 0 , δ 1 = -λ -N λ 1 , δ 0 = -N λ 0 , δ 1 = -N λ 1 , f j (t), j = 1, N -1, N g 0 (t) + f 0 (t),

Figure 2 :

 2 Figure 1. Approximated solution

  .49) Similarly, by (2.25), we deduce from (2.49) 1 , that|u m | p-2 u m → |u| p-2 u strongly in L 2 (Q T * ).

	(2.50)
	Passing to the limit in (2.37) by (2.47) -(2.50), we have u satisfying the problem
	

On the other hand -h 0 (t)u (0, t) u ′ (1, t) ≤

< 1, (4.12)

then I(t) > 0, ∀t ≥ 0.

Proof of Lemma 4.3. By the continuity of I(t) and I(0) > 0, there exists T 1 > 0 such that

this implies

where

Combining (4.7), (4.16) and using the Gronwal's inequality we have

where r as in (4.5). Hence, it follows from (4.12), (4.17) that

, by the continuity of I(t), we have I(T * ) ≥ 0. By the same arguments as in above part we can deduce that there exists

. Hence, we conclude that I(t) > 0, ∀t ≥ 0.

Lemma 4.3 is proved completely.

Lemma 4.4. Let I(0) > 0 and (4.13) hold. Then there exist the positive constants

for δ is small enough.

Proof of Lemma 4.4. It is easy to see that

Numerical results

Consider the following problem:

0 < x < 1, t > 0, with boundary conditions

and initial conditions u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), (5.3) where

are constants and the functions u 0 , u 1 , h 0 , h 1 , g 0 , g 1 and f are defined by

(5.4)

The exact solution of the problem (5.1) -( 5.3) with u 0 , u 1 , h 0 , h 1 , g 0 , g 1 and f defined in (5.4) respectively, is the function U ex given by U ex (x, t) = e x-t .

(

To solve problem (5.1) -(5.3) numerically, we consider the differential system for the unknowns U j (t) ≡ u(x j , t), V j (t) = dU j dt (t), with x j = j∆x, ∆x = 1 N , j = 0, 1, ..., N :

U j (0) = u 0 (x j ), V j (0) = u 1 (x j ), j = 0, N .

(5.6)

To solve the nonlinear differential system (5.6), we use the following linear recursive scheme gen-


MD Jonna Aaltonen

PhD Tiina Ojala

PhD Kirsi Laitinen

MSc Tuija Poussa

PhD Sue Ozanne

PhD Erika Isolauri

Impact of maternal diet during pregnancy and breastfeeding on infant metabolic programming: a prospective randomized controlled study

Keywords: Infant, Adiposity, Metabolism, Pregnancy, Counselling

Objectives: To evaluate the impact of maternal diet and intensive dietary counselling during pregnancy and breastfeeding on the infant's metabolic status.

Subjects/Methods: At the first trimester of pregnancy 256 women were randomized into a control/placebo group and two dietary counselling groups (diet/probiotics and diet/placebo). The counselling, with double-blind randomization to probiotics (Lactobacillus rhamnosus GG and Bifidobacterium lactis) or placebo, was targeting excessive saturated fat and low fibre consumption. The maternal diet was evaluated repeatedly during pregnancy and postpartum by means of three days' food diaries. Metabolic markers, serum 32-33 split and intact proinsulin, leptin/adiponectin ratio, skinfold thickness and waist circumference, were measured from 194 healthy infants at six months of age, and the high levels were taken to mirror adverse metabolic status.

Results:

The proportion of infants with high 32-33 split proinsulin was significantly lower in dietary counselling with probiotics(n=6/62, 9.7%) or placebo(n=7/69, 10.1%) compared to the control/placebo group(n=17/63, 27.0%). The high split proinsulin associated with larger skinfold thickness, waist circumference and higher leptin/adiponectin ratio in the infants(p<0.05). In respect of maternal diet during pregnancy, the highest and lowest tertiles of fat intake increased the infant's risk of high split proinsulin while those of butter associated correspondingly with the infants' waist circumference. Further, breastfed infants showed a reduced risk of high split proinsulin and leptin/adiponectin ratio compared to formula-fed infants.

Conclusions: Maternal diet during pregnancy and breastfeeding may benefit infant metabolic health. High split proinsulin reflects adverse metabolic status in infancy, which can be improved by early dietary counselling.

Introduction

Industrialized countries worldwide have been faced with a progressive increase in metabolic conditions such as obesity, and the velocity of propagation is particularly outstanding in the paediatric population. Evidence that overweight or obesity, heightened blood pressure and impaired glucose metabolism are programmed [START_REF] Barker | The developmental origins of chronic adult disease[END_REF]) by early nutrition points to foetal and early postnatal life as critical periods and intervention targets.

The data implicating early nutritional influences on the cardiometabolic risks in humans derive mainly from epidemiological studies of extreme prenatal circumstances such as exposure to famine [START_REF] Painter | Early onset of coronary artery disease after prenatal exposure to the Dutch famine[END_REF][START_REF] Ravelli | Glucose tolerance in adults after prenatal exposure to famine[END_REF]. In these demonstrations low birth weight is the main risk factor, taken to reflect a poor intrauterine nutritional environment. In well-nourished women, again, child birth weight, low or high, and a suboptimal intrauterine environment have been linked to maternal intake of single dietary factors [START_REF] Godfrey | Maternal nutrition in early and late pregnancy in relation to placental and fetal growth[END_REF][START_REF] Mikkelsen | Association between fruit and vegetable consumption and birth weight: a prospective study among 43,585 Danish women[END_REF][START_REF] Moses | Effect of a low-glycemic-index diet during pregnancy on obstetric outcomes[END_REF][START_REF] Zhang | Dietary fiber intake, dietary glycemic load, and the risk for gestational diabetes mellitus[END_REF]). However, a fact not previously properly addressed but not to be ignored is that the diet is always a mixture of several nutrients with possible complex interactions.

Previous demonstrations suggest that the gut microbiota composition is linked to the immunological and metabolic development of the child [START_REF] Guarner | Gut flora in health and disease[END_REF][START_REF] Rinne | Effect of probiotics and breastfeeding on the bifidobacterium and lactobacillus/enterococcus microbiota and humoral immune responses[END_REF]. The current intervention study targeted at dietary elements associated with these developmental components, specifically at the dietary low fibre and high saturated fat composition together with aberrant gut microbiota development by probiotics. We have shown that the joint actions of diet and probiotics benefited mothers' glucose metabolism [START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF]) and weight management (Ilmonen et al. Submitted).

Here, we studied, whether these benefits achieved in mothers' diet can be extended to their children's metabolic health at the age of six months. The high 32-33 split proinsulin, a well characterized predictor of insulin resistance in adults and older children [START_REF] Mykkanen | The relation of proinsulin, insulin, and proinsulin-to-insulin ratio to insulin sensitivity and acute insulin response in normoglycemic subjects[END_REF][START_REF] Singhal | Early nutrition and long-term cardiovascular health[END_REF][START_REF] Temple | Insulin deficiency in non-insulin-dependent diabetes[END_REF], was taken as a novel marker of adverse metabolic status in infancy. To further investigate the infants' metabolic status and the usefulness of high split proinsulin in its assessment, waist circumference, skinfold thickness and adipocyte-derived cytokines -leptin and adiponectin - [START_REF] Corvalan | Obesity indicators and cardiometabolic status in 4-y-old children[END_REF][START_REF] Darendeliler | Adiponectin is an indicator of insulin resistance in non-obese prepubertal children born large for gestational age (LGA) and is affected by birth weight[END_REF][START_REF] Shea | Obesity, fasting plasma insulin, and C-reactive protein levels in healthy children[END_REF][START_REF] Valle | Metabolic cardiovascular syndrome in obese prepubertal children: the role of high fasting insulin levels[END_REF] were taken as secondary outcomes.

Subjects and methods

Study design

A total of 256 pregnant women were recruited from the maternal welfare clinics in the area of Turku, Southwest Finland, in the first trimester of pregnancy between April 2002 and November 2004 to participate in a prospective, randomized mother-infant nutrition and probiotics study (NCT00167000; section 3, http://www.clinicaltrials.gov) [START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF]. Exclusion criteria for pregnant women were any chronic diseases except for allergy. The women visited the study clinic three times during pregnancy, at a median of 14 (range, 7 to 18), 24 (20 to 27) and 34 weeks of gestation (30 to 37), respectively. At the first study visit, representing the baseline, mothers were randomized into three study groups (Fig 1), two dietary intervention and one control group, according to computer-generated block randomization of six women. The study statistician, who was not involved in the study visits, generated the randomization list. The mother-infant pairs visited the study clinic at a median infant age of six months (range, 5 to 8).

The study complies with the Declaration of Helsinki, as revised in 2000. Written informed consent was obtained from the women and the Ethics Committee of the Hospital District of Southwest Finland approved the study.

Maternal dietary and probiotic intervention

The pregnant women in the control group received standard dietary counselling in well-women clinics. At each study visit, women in the dietary intervention groups (diet/probiotics and diet/placebo) received intensive dietary counselling in accordance with that currently recommended (1996), and this being additional to standard counselling given to all pregnant women in well-women clinics in Finland. The counselling was given in layman terms by a nutritionist who encouraged the participants to pay attention to the amount and type of fat and the amount of fibre in the diet [START_REF] Piirainen | Impact of dietary counselling on nutrient intake during pregnancy: a prospective cohort study[END_REF]. Further, to strengthen the dietary course and to demonstrate sources of favourable fat and fibre content, various food products, available on the market, were provided for use at home. The efficacy and safety of this dietary counselling during pregnancy have been reported elsewhere [START_REF] Piirainen | Impact of dietary counselling on nutrient intake during pregnancy: a prospective cohort study[END_REF]. The dietary intervention groups (diet/probiotics and diet/placebo) received capsules of probiotics (Lactobacillus rhamnosus GG, American type culture collection 53103, Valio Ltd., Helsinki, Finland and Bifidobacterium lactis, Chr. Hansen, Hoersholm, Denmark, 10 10 colony-forming units each/day) or placebo (microcrystalline cellulose and dextrose anhydrate; Chr. Hansen, Hoersholm, Denmark) in double-blind manner, while the control group (control/placebo) received placebo in single-blind manner. The maternal probiotic intervention continued from early pregnancy until the end of exclusive breastfeeding, a maximum six months postpartum.

Evaluation of maternal food and nutrient intake

The maternal dietary intake was evaluated in the context of all study visits by means of three days' food diaries, including one weekend day, and using household measures. The daily dietary intakes of energy, foods and nutrients were calculated by the computerized program Micro-Nutrica, version 2.5 (Research Centre of the Social Insurance Institution, Turku, Finland).

Maternal energy, fibre and energy-yielding nutrient intake was analyzed. The foods consumed were combined as groups (grain, meat, fish and dairy products, fruits and berries, soft margarine and vegetable oil, sugar and sweets) in the analyses, but milk, cheese, sour milk products, vegetables and butter consumptions were also analyzed separately.

Evaluation of clinical characteristics

Women's weight, height, blood pressure and fasting plasma glucose concentration were measured at every study visit (glucose not in the second visit) [START_REF] Aaltonen | Evidence of infant blood pressure programming by maternal nutrition during pregnancy: a prospective randomized controlled intervention study[END_REF][START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF]. Total gestational weight gain was calculated and the appropriate gestational weight gain evaluated according to the pre-pregnancy body mass index (1990). We recorded maternal diagnoses, including gestational diabetes mellitus, from the well-women records, and asked their smoking habits. The duration of pregnancy was calculated from the last menstruation.

Infants' weight, length and head circumference at birth were measured in the hospital maternity ward with measurements comparable to those used in the study visits. At the six months of age they underwent a physical examination, breastfeeding status was recorded, and their anthropometrics were measured; weight with Data baby scale 930 (Oriola, Espoo, Finland), length with Infantometer (Pedihealth, Oulu, Finland), waist and head circumference with a measuring tape, and the supra-iliac skinfold with a Holtain Tanner/Whitehouse Skinfold caliper (Marsden Weighing Group, Henley-on-Thames, Oxfordshire, UK).

Sampling

To evaluate infants' metabolic status, serum 32-33 split proinsulin, intact proinsulin and adiposity-derived hormones, leptin and adiponectin were chosen as metabolic markers, as these are not sensitive to non-fasting state [START_REF] Gil-Campos | Fasting and postprandial relationships among plasma leptin, ghrelin, and insulin in prepubertal obese children[END_REF][START_REF] Glauber | In vivo deactivation of proinsulin action on glucose disposal and hepatic glucose production in normal man[END_REF][START_REF] Karlsson | No postprandial increase of plasma adiponectin in obese subjects[END_REF]) unlike blood glucose and insulin concentrations. Indeed, for obvious ethical reasons overnight fasting was not possible at the age of six months. Venous blood samples were collected before noon and were successfully obtained from 194 (76%) infants (Fig 1).

Analytical methods

Serum was separated immediately and the samples were initially stored at -20°C and then at -70°C. The NIHR Cambridge Biomedical Research Centre, Core Biochemical Assay Laboratory (Cambridge, UK) analyzed the infants' samples. Serum leptin, adiponectin, 32-33 split proinsulin and intact proinsulin concentrations were assayed on a 1235 AutoDELFIA immunoassay system (Perkin Elmer Life Sciences, Boston, MA). All assays were in-house, two-step time-resolved fluorometric assays as previously described [START_REF] Hales | Fetal and infant growth and impaired glucose tolerance at age 64[END_REF][START_REF] Semple | Elevated plasma adiponectin in humans with genetically defective insulin receptors[END_REF][START_REF] Temple | Insulin deficiency in non-insulin-dependent diabetes[END_REF], and samples were analyzed in duplicate. Samples in which the coefficient of variation of the duplicates was greater than 10% were repeated. Quality Control samples with concentrations spanning the working range of the assay were run each day. The between-batch imprecision for the Quality Control samples was less than 8% for all assays and analyte concentrations. Maternal plasma glucose concentrations were analyzed on the day of sampling as previously reported [START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF].

Statistical analyses

The serum 32-33 split and intact proinsulin concentrations were primary outcome variables, leptin and adiponectin concentrations, waist circumference and supra iliac skinfold were secondary outcome variables. Any 32-33 split and intact proinsulin values above the 85 th percentile of the concentrations (7.9pmol/l and 6.64pmol/l, respectively) were considered high values. These were dichotomized, as the clinically important difference in mean levels is not known and we considered differences in proportions of higher values to be more relevant than those in overall mean levels. As the agreement between high concentrations was good (kappacoefficient κ= 0.80), only the dichotomized 32-33 split proinsulin concentration was analyzed as a final outcome variable. To evaluate high split proinsulin as a metabolic marker logistic regression analysis was used for dichotomized split proinsulin which was explained by continuous or dichotomized (the 85 th percentile as a cut-off point) adipocytokine ratio and anthropometrics measured at age of six months.

The group comparisons in categorized and continuous outcome variables were assessed by univariate logistic regression analysis or by analysis of variance (ANOVA) when appropriate. As most of the associations between the clinical characteristics (Table 1) and high split proinsulin were nonlinear, the characteristics of mother and child were categorized according to median, tertiles or quartiles. The effects of other possible explaining factors (Table 1) on infants' high 32-33 split proinsulin were analyzed using univariate logistic regression analyses. In the final multivariate models the intervention was forced and the explaining factors, if p< 0.10 in univariate analysis, were introduced to the forward stepwise logistic model (criterion for entry p< 0.10). The group comparisons are given as unadjusted and adjusted odds ratios (OR) with 95% confidence intervals using the control/placebo group as a reference group.

Maternal dietary intakes during and after pregnancy were divided into tertiles (T1=lowest, T2=middle and T3=highest). The Chi-squared test was used to study associations between dietary intakes and outcome variables. The effect of probiotic intervention on the association between dietary components and infant's high split proinsulin was analyzed by the method of Mantel-Haenszel and the Beslow-Day test, p<0.20 was used to indicate interaction. P-values < 0.05 were considered statistically significant. Statistical analyses were performed with SPSS version 15.0 (SPSS Inc. Chicago, IL, USA).

Results

Clinical characteristics of mothers and infants are shown in Table 1. All pregnant women were Caucasian, and the majority had college or university education (74%). The infants were healthy and their anthropometrics were comparable in the diet/probiotics, diet/placebo and control/placebo groups. The infants' median 32-33 split proinsulin concentration was 3.3 pmol/l (range: 1.2 to 21.9), intact proinsulin 3.7pmol/l (1.2 to 20.0), leptin 3.7ng/ml (0.7 to 14.8) and adiponectin 13.5 ug/ml (3.6 to 37.3) at age of six months.

The cluster of metabolic markers in infancy

In evaluating the usefulness of high split proinsulin as a marker of adverse metabolic status in infancy we found that higher skinfold thickness (OR=1.24, p=0.045), waist circumference (OR=1.14, p=0.035) and leptin/adiponectin ratio (OR=1.90, p=0.029), each of them included as continuous variables, were associated with increased likelihood of high 32-33 split proinsulin. Lower (<3320g; Q1 vs Q2-Q4) birth weight infants had an increased risk for subsequent high split proinsulin (OR=2.59, p=0.027), while the other infants' anthropometrics at birth or at the age of six months were not statistically significantly related to the high split proinsulin.

Impact of maternal dietary intervention on the infants' metabolic status

The infants' risk of high 32-33 split proinsulin concentration was lower in dietary counselling with probiotics or placebo groups compared to the control group (Fig 2 ); diet/probiotics unadjusted OR=0.29 (95% CI 0.11 to 0.80, p=0.016) and diet/placebo OR=0.31 (0.12 to 0.80, p=0.015). To find the difference between diet/probiotics (9.7%) and diet/placebo group (10.1%) with a 0.05 two-sided significance level and 90% power, the required sample size were more than 100 000 infants per group. The independent effect of dietary intervention on the infants' high split proinsulin concentration remained statistically significant in multivariate analysis with breastfeeding at six months, mother's glucose concentration at six months and weight gain during pregnancy. These adjusted and unadjusted logistic regression analyses are shown in Table 2.

According to univariate analyses the intervention did not affect other metabolic markers measured in the infants (data not shown), but the breastfed infants tended to have a reduced risk of high leptin/adiponectin ratio compared to formula-fed infants (OR 0.54; p=0.054).

Impact of diet on the infants' metabolic status

To evaluate the impact of maternal nutrition during pregnancy and at one and six months postpartum on the infants' risk of adverse metabolic programming, by means of high 32-33 split proinsulin, skinfold thickness, waist circumference and leptin/diponectin ratio, maternal dietary consumption was investigated in tertiles, as depicted in Table 3. Infants whose mothers' intake of fat, cheese or soft margarines and vegetable oil was in the highest or in the lowest tertile were more prone to the high split proinsulin than those whose mothers' intake was in the middle tertile. A corresponding non-linear association was found between maternal butter intake and infants' waist circumference and a tendency between maternal fat intake and infants' waist circumference. Extreme consumption of fruits and berries, butter and milk in the maternal diet resulted in contrast in lower infant risk of high split proinsulin compared to the middle tertile.

The highest tertile of grain products intake in the maternal diet was associated with the greatest risk of high split proinsulin and high skinfold thickness in the infants.

Interaction between probiotics and dietary components -exploratory analysis

To take into account the complexity of the diet, the interaction between probiotics and dietary components were studied. The effect of probiotic versus placebo administration on the percentage of infants with high 32-33 split proinsulin in tertiles of intake of dietary components are shown in Table 4. The probiotic intake contributed to the pattern of association between dietary components and infants' high split proinsulin according to the Breslow-Day test. The interaction was suggested if p<0.200. Interestingly, even when Bonferroni-adjusted, the maternal high milk, fruits and berries consumption combined with probiotics culminated in more prevalent high split proinsulin concentration in this group of infants compared to children whose mothers were not receiving probiotics. Further, the probiotics tended to reduced the detrimental effect of maternal low cheese, middle milk and low or middle fruit and berries consumption as well as high fat intake on the infant's risk of high split proinsulin concentration.

Discussion

Our results indicate that dietary counselling and a balanced nutritional environment early in life support a beneficial metabolic development of the infant. Together with our previous demonstration that maternal nutrition during pregnancy contributes to the infant's blood pressure [START_REF] Aaltonen | Evidence of infant blood pressure programming by maternal nutrition during pregnancy: a prospective randomized controlled intervention study[END_REF], the present findings would further support the conception that several risk factors in the metabolic syndrome may be modifiable by diet during critical and sensitive periods of life, as previously shown in adults [START_REF] Shiell | Diet in late pregnancy and glucoseinsulin metabolism of the offspring 40 years later[END_REF].

From our prospective study we have thus far learned that by modifying particularly the dietary intake of fat and fibre, by detailed dietary counselling and the provision of appropriate food products combined with probiotics, we are not only improving the quality of the maternal diet [START_REF] Piirainen | Impact of dietary counselling on nutrient intake during pregnancy: a prospective cohort study[END_REF]) but also the maternal glucose metabolism up to one year after pregnancy [START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF]). In the present study the risk reduction in infants' high 32-33 split proinsulin concentration was already detected at six months of age, although, the intervention did not influence the infants' adiposity measurements or leptin/adiponectin ratio. Further, maternal dietary counselling per se, with or without probiotics, was independently related to a lowered risk of high split proinsulin concentration in infants irrespective of possible confounding variables such as maternal gestational diabetes mellitus, pregnancy weight gain or child birth size, previously linked to adverse metabolic outcomes in the child [START_REF] Boney | Metabolic syndrome in childhood: association with birth weight, maternal obesity, and gestational diabetes mellitus[END_REF][START_REF] Wrotniak | Gestational weight gain and risk of overweight in the offspring at age 7 y in a multicenter, multiethnic cohort study[END_REF].

Since the maternal and foetal nutritional environments are closely related, an explanation for the beneficial effect of dietary counselling could be extrapolated from the current knowledge that a higher intake of unsaturated fatty acids can improve insulin sensitivity [START_REF] Riserus | Dietary fats and prevention of type 2 diabetes[END_REF], whereas a high saturated fat content in the diet promotes the secretion of pro-inflammatory cytokines [START_REF] Cani | Selective increases of bifidobacteria in gut microflora improve high-fat-diet-induced diabetes in mice through a mechanism associated with endotoxaemia[END_REF], causally linked to insulin resistance [START_REF] Shoelson | Inflammation and insulin resistance[END_REF]. Further, higher fibre consumption, by reducing the risk of gestational diabetes mellitus [START_REF] Zhang | Dietary fiber intake, dietary glycemic load, and the risk for gestational diabetes mellitus[END_REF] or by facilitating maternal weight control [START_REF] Howarth | Dietary fiber and weight regulation[END_REF], was presupposed to benefit the infant's metabolic programming. These previous data are in agreement with our findings that maternal intake of fat and specific fat-containing food products affected metabolic markers in infancy. The impacts of grain products, fruits and berries on the infants' risk may be mediated by the effect of dietary fibre. Interestingly, in contrast to what has previously been shown [START_REF] Godfrey | Maternal nutrition in early and late pregnancy in relation to placental and fetal growth[END_REF], maternal energy, protein or carbohydrate intakes were not related to infants' high split proinsulin concentration in our analyses. Furthermore, a longer duration of breastfeeding reduced the infants' risk of high split proinsulin and leptin/adiponectin ratio. The advantage of breastfeeding may be related to protection against obesity [START_REF] Armstrong | Breastfeeding and lowering the risk of childhood obesity[END_REF] or to the collective composition of the gut microbiota. Indeed, bifidobacteria, which typify the gut microbiota of the healthy breast-fed infant [START_REF] Fanaro | Intestinal microflora in early infancy: composition and development[END_REF], may dampen the systemic endotoxemia induced by bacterial lipopolysaccarides [START_REF] Griffiths | In vivo effects of bifidobacteria and lactoferrin on gut endotoxin concentration and mucosal immunity in Balb/c mice[END_REF], and thereby improve metabolic status [START_REF] Cani | Selective increases of bifidobacteria in gut microflora improve high-fat-diet-induced diabetes in mice through a mechanism associated with endotoxaemia[END_REF][START_REF] Shoelson | Inflammation and insulin resistance[END_REF]. The independent effect of maternal probiotic intervention on the infant's risk of high split proinsulin was not possible to study in this study population as the estimated sample size for such evaluation were more than 100 000 mother-infant pairs per group. However, our exploratory results suggest that probiotics may interact with dietary components and enhance the association between maternal dietary components and infants' high split proinsulin concentration. In the light of previous studies this is not surprising. For example, dietary fatty acids and gut microbiota share similar signalling pathways in immune responses possibly controlling the low-grade inflammation [START_REF] Laitinen | Breast milk fatty acids may link innate and adaptive immune regulation: analysis of soluble CD14, prostaglandin E2, and fatty acids[END_REF]) frequently detected in metabolic disorders [START_REF] Fantuzzi | Adipose tissue, adipokines, and inflammation[END_REF]. Thus our results, together with those of recent human and animal studies, point to the relevance of late gestation and early postnatal life [START_REF] Jones | Fetal programming of glucose-insulin metabolism[END_REF], and the importance of dietary quality [START_REF] Cani | Selective increases of bifidobacteria in gut microflora improve high-fat-diet-induced diabetes in mice through a mechanism associated with endotoxaemia[END_REF] in the metabolic development.

A large body of evidence supports the role of 32-33 split proinsulin as a marker of insulin resistance. Raised concentrations of split proinsulin in adults have been interpreted as evidence of beta-cell dysfunction [START_REF] Temple | Insulin deficiency in non-insulin-dependent diabetes[END_REF]) and a risk of impaired glucose tolerance [START_REF] Hales | Fetal and infant growth and impaired glucose tolerance at age 64[END_REF], but its role as a metabolic marker in infancy has remained poorly understood [START_REF] Hawdon | The role of pancreatic insulin secretion in neonatal glucoregulation. I. Healthy term and preterm infants[END_REF][START_REF] Singhal | Early nutrition and long-term cardiovascular health[END_REF]). We found a positive association between the high split proinsulin concentration and the ratio of adipocyte-derived cytokines -leptin and adiponectin -which have been found to correlate with adiposity [START_REF] Mantzoros | Cord blood leptin and adiponectin as predictors of adiposity in children at 3 years of age: a prospective cohort study[END_REF][START_REF] Schubring | Leptin serum concentrations in healthy neonates within the first week of life: relation to insulin and growth hormone levels, skinfold thickness, body mass index and weight[END_REF], and metabolic disorders [START_REF] Darendeliler | Adiponectin is an indicator of insulin resistance in non-obese prepubertal children born large for gestational age (LGA) and is affected by birth weight[END_REF][START_REF] Valle | Metabolic cardiovascular syndrome in obese prepubertal children: the role of high fasting insulin levels[END_REF] in newborns and children, and, further, to reflect efficaciously cardio-metabolic risks [START_REF] Norata | Leptin:adiponectin ratio is an independent predictor of intima media thickness of the common carotid artery[END_REF][START_REF] Steinberger | Relation of leptin to insulin resistance syndrome in children[END_REF]. On the other hand, weight status at the age of six months predicts obesity in childhood [START_REF] Taveras | Weight status in the first 6 months of life and obesity at 3 years of age[END_REF], and here abdominal obesity, known to impair beta-cell function [START_REF] Hanley | Cross-sectional and prospective associations between abdominal adiposity and proinsulin concentration[END_REF], was linked to the high split proinsulin concentration. A limitation to acknowledge is that the infant were not fasted before blood sampling. As this being unethical we selected metabolic markers not very sensitive to non-fasting state [START_REF] Gil-Campos | Fasting and postprandial relationships among plasma leptin, ghrelin, and insulin in prepubertal obese children[END_REF][START_REF] Glauber | In vivo deactivation of proinsulin action on glucose disposal and hepatic glucose production in normal man[END_REF][START_REF] Karlsson | No postprandial increase of plasma adiponectin in obese subjects[END_REF]. Taken together, a high level of split proinsulin at the age of six months likely indicates susceptibility to adverse metabolic programming in these infants.

In conclusion, we have shown for the first time in humans that favourable metabolic programming, measured especially by a lower incidence of high 32-33 split proinsulin, can be achieved by balancing the diet of the mother during pregnancy and by breastfeeding the infant. Thus, the intrauterine and immediate postnatal period comprises a window of opportunity for interventions aiming to reduce the risk of metabolic disorders in both mother and infant and implies a prospect of achieving health benefits for two generations.

Figure 2

The impact of dietary intervention on the infants' high (above the 85 Results are given as means (SD) or medians (range) or number of subjects (percentage). The statistically significant differences in mothers among the study groups have been published elsewhere [START_REF] Laitinen | Probiotics and dietary counselling contribute to glucose regulation during and after pregnancy: a randomised controlled trial[END_REF], Luoto et al. 2010, Ilmonen et al. Submitted).

a BMI = body mass index

The groups were compared using ANOVA unless stated otherwise.

b Chi-squared test, c ANCOVA for repeated measurements with 1 st trimester as a covariate, d Kruskal-Wallis test, e ANOVA for repeated measures.

Figure legends Figure 1

 1 Figure legends

Table 1

 1 Clinical characteristics of mothers and their children (n=194) in the dietary intervention groups (diet/probiotics and diet/placebo) and in the control group (control/placebo)

	th percentile

Table 2 .

 2 The effect of interventions and clinical characteristics on infant's high 32-33 split proinsulin (above the 85 th percentile cut-off point) in univariate and multivariate analysis.

	Explaining factors	Univariate analysis	Multivariate analysis a
					Adjusted
		OR (95% CI)	P-value	OR (95% CI)	P-value
	Intervention b
	Control/placebo (n=63)	1.00		1.00
	Diet/probiotics (n=62)	0.29 (0.11 to 0.80)	0.016	0.24 (0.07 to 0.79)	0.019
	Diet/placebo (n=69)	0.31 (0.12 to 0.80)	0.015	0.27 (0.09 to 0.79)	0.017
	Breastfeeding at 6 months
	Ended (n=57)	1.00		1.00
	Exclusive or partial (n=136) 0.21 (0.09 to 0.47)	<0.001 0.24 (0.09 to 0.60)	0.002
	Glucose 6 mo postpartum
	< 4.90mmol/l (n=90)	1.00		1.00
	≥ 4.90mmol/l (n=104)	3.37 (1.37 to 8.28)	0.008	3.06 (1.10 to 8.50)	0.032
	Weight gain during pregnancy c
	As recommended (n=77)	1.00		1.00
	> recommended (n=80)	2.12 (0.81 to 5.58)	0.128	2.77 (0.92 to 8.30)	0.070
	< recommended (n=32)	3.33 (1.09 to 10.17)	0.034	4.04 (1.13 to 14.46)	0.032
	The results are given as non-adjusted odds ratios (95% CI) and as adjusted for clinical characteristics
	using multivariable logistic regression analysis. Only explaining factors included in the final
	multivariate model of infants' high 32-33 split proinsulin concentration are shown.

a The intervention was forced to the model and the following categorical variables were given to the

Table 3

 3 Tertiles for intake of foods and energy-yielding nutrients in maternal diet during pregnancy and breastfeeding, and the proportion of infants with high (above the 85 th percentile cut-off point) 32-33 split proinsulin concentration, skinfold thickness, waist circumference and leptin/adiponectin ratio in each tertile of intake.Sample available for infant 32-33 split proinsulin analysis at age six months n= 194

	Dietary	Study	Tertile of intake	Infants with high	Infants with high	Infants with high	Infants with high
	component	Visit				split proinsulin	skinfold thickness	waist circumference	leptin/adiponectin
				Mean	Range	Number	P a	Number	P a	Number	P a	Number	P a
						(%)		(%)		(%)		(%)
	Grain products, g 1 st trim. T1 (n=65) 134	43-176	7 (10.8) 0.034 11 (18.6) 0.011	7 (11.7)	0.065	10 (15.4) 0.491
			T2 (n=70) 206	176-239	8 (11.4)		3 (4.5)		5 (7.8)		8 (11.4)
			T3 (n=58) 298	239-560	15 (25.9)		12 (23.1)		12 (22.2)		11 (19.0)
	Fruits and	2 nd trim. T1 (n=69) 147	17-232	6 (8.7)	0.006	8 (12.3)	0.518	8 (12.5)	0.597	12 (17.4)	0.571
	berries, g		T2 (n=62) 300	233-373 17 (27.4)		7 (12.7)		10 (16.9)		10 (16.1)
			T3 (n=63) 543	373-1443 7 (11.1)		11 (19.0)		6 (10.7)		7 (11.1)
	Fat, g	3 rd trim. T1 (n=59)	46	23-58	11 (18.6) 0.038	8 (14.8)	0.243	7 (12.3)	0.088	8 (13.6)	0.234
			T2 (n=66)	67	58-76	4 (6.1)		12 (20.3)		4 (7.0)		7 (10.6)

Acknowledgements

We thank the NIHR Biomedical Research Centre Cambridge and Core Biochemistry Assay Laboratory for proinsulin, leptin and adiponectin measurements, Dr Anu Huurre and Ms Ulla-Maija Eriksson for their contribution in contacting the study subjects, Ms Satu Leinonen for assistance in sample procedures and Mr Robert MacGilleon for revision of the English text. This study was supported by grants from the Social Insurance Institution of Finland, the Sigrid Juselius Foundation, the Yrjö Jahnsson Foundation, the Juho Vainio Foundation (JA), the Varsinais-Suomi Regional Fund of the Finnish Cultural Foundation (JA) and the Emil Aaltonen Foundation (JA). Food products were donated by Raisio plc (Raisio, Finland), L. rhamnosus GG by Valio Ltd (Helsinki, Finland) and B. lactis by Chr. Hansen (Hoersholm, Denmark).

This study was supported by grants from the Social Insurance Institution of Finland, the Sigrid Juselius Foundation, the Yrjö Jahnsson Foundation, the Juho Vainio Foundation (JA), the Varsinais-Suomi Regional Fund of the Finnish Cultural Foundation (JA) and the Emil Aaltonen Foundation (JA). Food products were donated by Raisio plc (Raisio, Finland), L. rhamnosus GG by Valio Ltd (Helsinki, Finland) and B. lactis by Chr. Hansen (Hoersholm, Denmark).

Conflict of interest:

The authors declare no conflict of interest.

Only dietary components significantly associated with infants' high 32-33 split proinsulin concentration are shown.

a The association between tertiled intake of dietary components and infants' high split proinsulin concentration, skinfold thickness, waist circumference and leptin/adiponectin ratio were analyzed by Chi-squared test in combined study groups (diet/probiotics, diet/placebo and control/placebo).