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

Abstract

This paper deals with u t = Δ u + u m ( x, t) e pv(0, t) , v t = Δ v + u q (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 u 0( v 0) 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.