Necessary and Sufficient Condition for Global Existence for a Degenerate Parabolic Boundary Value Problem

Consider the degenerate parabolic boundary value problem utsDw u.qf u. on V= 0, `.in which V is a bounded domain in RNand the C w0, `.. functions f and f are nonnegative and nondecreasing with w s.f s.)0 if s)0 and w 0.s0. Assume homogeneous Neumann boundary conditions and an initial condition that is nonnegative, nontrivial, and continuous on V. Because the function w 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 w Appl. Math. Optim. 17 1988., 203]224x. We show that this initial boundary value problem has such a nonnegative generalized solution if and only if H0` dsr 1qf s.. s`.