Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

Abstract

This paper deals with ut = Δu + um(x, t)e pv(0,t), vt = Δv + uq(0, t)e nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u0(v0) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ≤ m ≤ 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions. © 2010 Elsevier Inc. All rights reserved.