On Blow-up Solutions of A Parabolic System Coupled in Both Equations and Boundary Conditions

Abstract

This paper is concerned with the blow-up solutions of a system of two reaction-diffusion equations coupled in both equations and boundary conditions. In order to understand how the reaction terms and the boundary terms affect the blow-up properties, the lower and upper blow-up rate estimates are derived. Moreover, the blow-up set under some restricted assumptions is studied.

قابلية الحل للسيطرة الامثلية الحدودية الرباعية المستمرة التقليدية المسيطر بالنظام الرباعي المكافيء

“…The study of the continuous classical type began with the continuous classical boundary optimal control problems dominated by nonlinear parabolic or elliptic or hyperbolic PDEs. Then, these studies were generalized to deal with systems dominated by coupling nonlinear PDEs of these three types 10,11 , and then were generalized also to deal systems dominated by triple nonlinear PDEs of these three types 12 . In each type of these classical continuous boundary optimal control problems, the problem consists of; an initial or a boundary value problem (the dominating eqs.…”

https://www.isr-publications.com/jmcs/articles-12886-numerical-finite-difference-approximations-of-a-coupled-parabolic-system-with-blow-up

“…In general, [19], for a time-dependent equation,defined in Ω ⊂ R n , one can say that the classical solution u(x, t) blows up in L ∞ -norm or blows up (for short), if there exists T < ∞, called the blow-up time, such that u is well defined for all 0 < t < T , while it becomes unbounded in L ∞ -norm, when t approaches to T , that is: sup…”