This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a p-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Holder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller–Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.
Concerning uniqueness of solution, the presence of a nonlinear degenerate diffusion term and a nonlinear transport term represents a disadvantage and we could not obtain the uniqueness of a weak solution. This contrasts with the results by @cite_20 , where the authors prove uniqueness of solutions for a degenerate parabolic-elliptic system set in an unbounded domain, using a method which relies on a continuous dependence estimate from @cite_5 , that does not apply to our problem because it is difficult to bound @math in @math due to the parabolic nature of .
@cite_5 @cite_20
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We study nonlinear degenerate parabolic equations where the flux function @math does not depend Lipschitz continuously on the spatial location @math . By properly adapting the "doubling of variables" device due to Kružkov [25] and Carrillo [12], we prove a uniqueness result within the class of entropy solutions for the initial value problem. We also prove a result concerning the continuous dependence on the initial data and the flux function for degenerate parabolic equations with flux function of the form @math , where @math is a vector-valued function and @math is a scalar function. This paper considers a trapped characteristic initial value problem for the spherically symmetric Einstein-Maxwell-scalar field equations. For an open set of initial data whose closure contains in particular Reissner-Nordstrdata, the future boundary of the maximal domain of development is found to be a light-like surface along which the curvature blows up, and yet the metric can be continuously extended beyond it. This result is related to the strong cosmic censorship conjecture of Roger Penrose. The principle of determinism in classical physics is expressed mathemat- ically by the uniqueness of solutions to the initial value problem for certain equations of evolution. Indeed, in the context of the Einstein equations of general relativity, where the unknown is the very structure of space and time, uniqueness is equivalent on a fundamental level to the validity of this principle. The question of uniqueness may thus be termed the issue of the predictability of the equation. The present paper explores the issue of predictability in general relativity. Since the work of Leray, it has been known that for the Einstein equations, contrary to common experience, uniqueness for the Cauchy problem in the large does not generally hold even within the class of smooth solutions. In other words, uniqueness may fail without any loss in regularity; such failure is thus a global phenomenon. The central question is whether this violation of predictability may occur in solutions representing actual physical processes. Physical phenomena and concepts related to the general theory of relativity, namely gravitational collapse, black holes, angular momentum, etc., must cer- tainly come into play in the study of this problem. Unfortunately, the math- ematical analysis of this exciting problem is very difficult, at present beyond reach for the vacuum Einstein equations in the physical dimension. Conse- A new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation. By extending the theory of viscosity solutions, a comparison principle is established. For periodic continuous initial data a unique global continuous solution (periodic in space) is constructed. The theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions. Even if the driving force term (homogeneous in space) exists, the initial-value problem is solvable for general nonadmissible continuous (periodic) initial data. Abstract In [18] Paicu and Zarnescu have studied an order tensor system which describes the flow of a liquid crystal. They have proven the existence of weak solutions, the propagation of higher regularity, namely H s with s > 1 and the weak-strong uniqueness in dimension two. This paper is devoted to fill the gap of their results, namely to propagate the low regularity, namely H s for 0 s 1 and to prove the uniqueness of the weak solutions. For the completeness of this research, we also propose an alternative approach in order to prove the existence of weak solutions.