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

We study positive blowing-up solutions of the system: u(t)-delta Delta u = v(P), v(t)-Delta v = u(q) , 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 delta > 0 is allowed; (iii) a large class of nonlinearities F(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. (C) 2015 Elsevier Inc. All rights reserved.