121326094
On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary Condition
This paper deals with solutions $u(t,x)$ of parabolic differential inequalities (a) $u_1 \leqq Lu$, or (b) $u_1 \geqq Lu$, respectively, where L is a linear, weakly elliptic differential operator of second order. The behavior of u for large t is studied under the assumption, that on the lateral boundary a nonlinear boundary condition of the form (a) ${{\partial u} / {\partial v}} \leqq f(u)$, or (b) ${{\partial u} / {\partial v}} \geqq f(u)$, is imposed, where $f(z) \to \infty $ as $z \to \infty $. It is shown that the value of the integral $\int^\infty {{dz} / {f(z)f'(z)}}$ is crucial for the growth properties of u. If this integral is infinite, then we have the case of global existence, i.e., any solution u of (a) is bounded in bounded sets. If, on the other hand, the integral is finite, then all solutions u of (b) with large initial values become infinite in finite time.
120951143
Verallgemeinerte Laplace-Operatoren und Potentiale
119881926
Fehlerabschätzungen und Eindeutigkeitssätze für gewöhnliche und partielle Differentialgleichungen
122666640
Remarks on a paper by F. Browder about contraction
THROUGHOUT this note, X denotes a complete metric space with distance function d, and T is a map from X into itself. Powers of T are defined by Tax = x and T”+ lx = T(T”x), n 2 0. Occasionally, we use the notation xk = Tkx, in particular x0 = x, x1 = TX, for the sake of brevity. The set {x, TX, T'x, . . . , T”x,. . .} is called an orbit (starting at x) and denoted by O(x). The set 0(x, y) is the union of the two orbits starting at x and y, 0(x, y) = {xk, yk: k 3 O}. The diameter diam A of a nonempty subset A of X is the supremum of the numbers d(x, y), where x and y vary in A. The letter 4 denotes a contractive gauge function, i.e. a continuous, increasing function from R,, the non-negative reals, into R, which satisfies 4(s) 0.
123298075
Eindeutigkeitssätze für gewöhnliche, parabolische und hyperbolische Differentialgleichungen
122832071
THERE IS AN ELEMENTARY PROOF OF PEANO'S EXISTENCE THEOREM.
Abstract : An elementary, constructive proof of Peano's existence theorem is given. The maximal solution is constructed as a limit of a decreasing sequence of approximate solutions. (Author)
1300115
Flexible precision control and dynamic data structures for programming mathematical and numerical algorithms