Critical exponents for non-simultaneous blow-up in a localized parabolic system Original Research Article

The paper deals with the radially symmetric solutions of ut=Δu+um(x,t)vn(0,t)ut=Δu+um(x,t)vn(0,t), vt=Δv+up(0,t)vq(x,t)vt=Δv+up(0,t)vq(x,t), subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that uu (vv) blows up alone if and only if m>p+1m>p+1 (q>n+1q>n+1), which means that any blow-up is simultaneous if and only if m≤p+1m≤p+1, q≤n+1q≤n+1. (ii) Any blow-up is uu (vv) blowing up with vv (uu) remaining bounded if and only if m>p+1m>p+1, q≤n+1q≤n+1 (m≤p+1m≤p+1, q>n+1q>n+1). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1m>p+1, q>n+1q>n+1. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.