A Necessary and Sufficient Condition for Global Existence for a Degenerate Parabolic Boundary Value Problem
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 ifi∞0 ds/(1 + f(s)) = ∞.
Journal of Mathematical Analysis and Applications
Lair, A. V, & Oxley, M. E. (1998). A Necessary and Sufficient Condition for Global Existence for a Degenerate Parabolic Boundary Value Problem. Journal of Mathematical Analysis and Applications, 221(1), 338–348. https://doi.org/10.1006/jmaa.1997.5900