Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values (Q1265207)

It is considered the initial value problem for the semilinear parabolic system \[ u_t=\Delta u+v^p,\quad v_t=\Delta v+u^q, \quad (x,t)\in \mathbb{R}^N\times \mathbb{R}^+. \] At \(t=0\), nonnegative, bounded and continuous initial values \((u_0(x),v_0(x))\) are prescribed. The main results are for the case when \(u_0 \sim (\lambda| x| ^{-\alpha})^{1/(q+1)}\), \(v_0 \sim (\lambda| x| ^{-\alpha})^{1/(p+1)}\) with \(\lambda>0\), \(0\leq a<N\min \{p+1,q+1\}\). The authors consider various questions of global existence and nonexistence, large time behavior or life span of the solutions in terms of simple conditions on \(\lambda, a, p, q\) and the space dimension \(N\).