A necessary and sufficient condition for global existence for a degenerate parabolic boundary value problem

摘要

Consider the degenerate parabolic boundary value problem u(t) = Delta phi(u) + f(u) on Omega X (0, infinity) in which Omega is a bounded domain in R-N and the C([0, infinity)) functions f and phi are nonnegative and nondecreasing with phi(s)f(s) > 0 if s > 0 and phi(0) = 0. Assume homogeneous Neumann boundary conditions and an initial condition that is nonnegative, nontrivial, and continuous on <(Omega)over bar>. Because the function phi is not sufficiently nice to allow this problem to have a classical solution, we consider generalized solutions in a manner similar to that of Benilan, Crandall, and Sacks [Appl. Math. Optim. 17 (1988), 203-224]. We show that this initial boundary value problem has such a nonnegative generalized solution if and only if integral(0)(infinity) ds/(1 + f(s)) = infinity. (C) 1998 Academic Press. [References: 17]