Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

Abstract

This paper deals with a class of localized and degenerate quasilinear parabolic systemswith homogeneous Dirichlet boundary conditions. Local existence of positive classical solutions is proven by using the method of regularization. Global existence and blow-up criteria are also obtained. Moreover, the authors prove that under certain conditions, the solutions have global blow-up property.

Some Types of Reaction-Diffusion Systems With Nonlocal Boundary Conditions

“…We also assume that (u 0 , v 0 ) satisfies the compatibility condition on ∂Ω, and that ϕ(x, ·) ≡ 0 and ψ(x, ·) ≡ 0 for any x ∈ ∂Ω for the sake of the meaning of nonlocal boundary conditions. Systems (1.1) and (1.3) describe some physical phenomena in which the nonlinear reaction in a dynamical system takes place only at a single point (see [3,13,17,24]). Systems (1.2) and (1.4) are related to some ignition models for compressible reactive gases (see [2,24]).…”

A degenerate parabolic system with localized sources and nonlocal boundary condition

“…For other related work with localized sources, we refer the reader to [8][9][10][11]23,28,29,49] and the references therein.…”