Title

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

Document Type

Article

Publication Date

5-1-1998

Abstract

Consider 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 ifi0 ds/(1 + f(s)) = ∞.

Comments

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Reviewed at MR1619148

DOI

10.1006/jmaa.1997.5900

Source Publication

Journal of Mathematical Analysis and Applications

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