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.