Improved conditions for single-point blow-up in reaction–diffusion systems

We study positive blowing-up solutions of the system:ut−δΔu=vp,vt−Δv=uq, as well as of some more general systems. For any p,q>1, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions:(i) no type I blow-up assumption is made (and it is known that this property may fail);(ii) no equidiffusivity is assumed, i.e. any δ>0δ>0 is allowed;(iii) a large class of nonlinearities F(u,v)F(u,v), G(u,v)G(u,v) can be handled, which need not follow a precise power behavior.As a side result, we also obtain lower pointwise estimates for the final blow-up profiles.