A Fujita‐type global existence—global non‐existence theorem for a system of reaction diffusion equations with differing diffusivities

摘要

AbstractIn this paper we condiser non‐negative solutions of the initial value problem in ℝNfor the systemdocumentclass{article}pagestyle{empty}begin{document}$$ {rm u}_{{rm t = }} {rm delta Delta u + v}^{rm p}, $$end{document}documentclass{article}pagestyle{empty}begin{document}$$ {rm v}_{{rm t = }} {rm Delta v + u}^{rm q}, $$end{document}where 0 ⩽ δ ⩽ 1 andpq>0. We prove the following conditions.Suppose min(p,q)≥1 butpq1.(a)If δ = 0 thenu=v=0 is the only non‐negative global solution of the system.(b)If δ>0, non‐negative non‐globle solutions always exist for suitable initial values.(c)If 02, 00, wherecdepends only upon the initial data.(e)Suppose 0>δ 1 and max (α, β)= 1,2 orN>2 and max (p,q)⩽N/(N‐2), then global, non‐trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert spaceH1(K) whereK(x)  exp(¼∣x∣2). They decay like emax(α,β)‐(N/2)+εtfor every ε>0. These solutions are classical solutions fort>0.(f)If max (α, β)1 and δ = 0, than all non‐trivial non‐negativemaximalsolutions are non‐global.(i)If 01 and max(α,β)≥N/2 all non‐trivial non‐negativemaximalsolutions are non‐global.(j)If 01 and max(α,β)