Non-simultaneous blow-up and blow-up rates for reaction–diffusion equations

This paper considers blow-up solutions for reaction–diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of nn components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of k(∈{1,2,…,n})k(∈{1,2,…,n}) components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of nn components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215–231].