Liouville theorems and blow up behaviour in semilinear reaction diffusion systems (Q1357496)

Positive solutions to the system \[ u_{t} = \Delta u+v^{p}, \quad p\geq 1,\qquad v_{t} = \Delta v+u^{q}, \quad q\geq 1,\tag{1} \] are studied, which blow up at \(x=0\) and \(t=T<\infty.\) The authors obtain conditions on \(p,\) \(q\) and the spatial dimension \(N,\) ensuring the following bounds for the blow up rates: \[ u(x,t)\leq C(T-t)^{-(p+1)/(pq-1)},\qquad v(x,t)\leq C(T-t)^{-(q+1)/(pq-1)} \tag{2} \] for some positive constant \(C.\) A complete classification of the blow up patterns is performed by virtue of \((2)\) and parabolic Liouville type theorems. Finally, existence of solutions to \((1)\) is proved which exhibits a type of asymptotics near blow up which is qualitatively different from those in the scalar case.