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

Abstract

We consider the semilinear parabolic system u_{t}=\triangle u+v^{p} , v_{t}=\triangle v+u^{q} , where x\in R^{N}(N\geq 1) , t>0 and p , q\geq 1 . 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 (u0, v0) have polynomial decay near x=\infty . Assuming u_{0}\sim(\lambda|x|^{-a})^{1/(q+1)} , v_{0}\sim(\lambda|x|^{-a})^{1/(p+1)} with \lambda >0,0 \leq a<N\min\{p+1, q+1\} , we answer 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 .