Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources

Abstract

This paper deals with two classes of parabolic systems with localized nonlinear sources. The critical exponents as well as the estimates for blow-up rates and boundary layer profiles are determined.  2004 Elsevier Inc. All rights reserved.

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

“…If a(x) = 1, these problems are those from various models in physics theory and engineering application, and have been studied by numerous authors. The details can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein. Among them, Souplet [3] introduced a method to investigate the profile of blow-up solutions of nonlinear diffusion equations with nonlocal reaction terms, and observed the asymptotic blow-up behaviors of the solutions of a large class of equations.…”

Global Existence and Blow-Up for a Degenerate Reaction-Diffusion System With Nonlinear Localized Sources and Nonlocal Boundary Conditions

“…In the past several decades, there have been many articles deal with properties of solutions to porous medium equations or degenerate parabolic system with a localized source subject to homogeneous Dirichlet boundary condition and to a system of heat equations with nonlinear boundary condition (see [5,9,11,13,21,25,26] and references therein). However, there are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modelling such as thermoelasticity theory (see [4,6,7]).…”

A degenerate parabolic system with localized sources and nonlocal boundary condition

“…The solution (u(x, t), v(x, t)) of problem (1. In recent years, the parabolic equations and systems with localized sources have been discussed by many authors, see [4,6,[25][26][27]32,36,40,41,48] and the references therein. Particularly, as a physical motivation, boundary value problem u t = Δ u m + u p u q (x 0 , t), x ∈ Ω, t > 0,…”

“…was considered in [27] and [48], respectively. The parameters p i , q i (i = 1, 2) are nonnegative numbers with p 2 q 1 > 0.…”