Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

Abstract

We concider, the blow-up solutions for a coupled reaction diffusion system with gradient terms. The main purpose is to understand whether the gradient terms effect the blow-up properties. We derive the upper and lower blow-up rate estimates under certain assumptions.

Some Blow-Up Properties of a Semilinear Heat Equation

“…The aim of paper, is to show that the results of Friedman and McLeod hold true for problem (1). In other words, we prove that = 0 is the only possible blow-up point for this problem.…”

“…The blow-up phenomena in parabolic problems, defined on bounded domains, have been studied by many authors, see for instance [1][2][3][4][5][6][7].…”

“…On the other hand, it has been shown in [4] that the upper (lower) blow-up rate estimates take the following form 1) , ∈ (0, ).…”

“…In other words, we prove that = 0 is the only possible blow-up point for this problem. Moreover, we derive the upper blow-up rate estimate for problem (1). The rest of this paper is organized as follows: In section two, we discuss the local existance and blow-up with stating some properties of classical solutions of problem (1).…”

Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

“…In some cases, the solutions of parabolic problems cannot be continued globally in time. This is called the blow-up phenomenon which, in time-dependent problems, has been studied over the past years by many authoes [3][4][5][6][7]. One of these problems is the problem of the heat equation defined in a ball with a nonlinear Neumann boundary condition, on which has been introduced in previous articles [7][8][9][10][11].…”