Large time behavior of ODE type solutions to a nonlinear parabolic system

Abstract

Let (u,v) be a solution to the Cauchy problem for a nonlinear parabolic system (P)∂tu=Δu+vαinRN×(0,∞),∂tv=Δv+uβinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN,v(x,0)=μ+ψ(x)>0inRN,where α, β>0 with αβ<1, λ,μ>0 and φ,ψ∈BC(RN)∩Lr(RN) with 1≤r<∞ and infx∈RNφ(x)>−λ, infx∈RNψ(x)>−μ. Then the solution (u,v) to problem (P) behaves like a positive solution to ODE’s ζ′=ηα and η′=ζβ in (0,∞) and both of ‖u(t)‖L∞(RN) and ‖v(t)‖L∞(RN) diverge as t→∞. In this paper we obtain the precise description of the large time behavior of the solution (u,v).