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

AbstractConsider the degenerate parabolic boundary value problemut=Δϕ(u)+f(u) on Ω×(0,∞) in which Ω is a bounded domain inRNand theC([0,∞)) functionsfand φ are nonnegative and nondecreasing with ϕ(s)f(s)>0 ifs>0 and ϕ(0)=0. Assume homogeneous Neumann boundary conditions and an initial condition that is nonnegative, nontrivial, and continuous on Ω. Because the function ϕ 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 ifi∞0ds/(1+f(s))=∞