Existence and nonexistence of global solutions for a semilinear reaction–diffusion system

This paper is concerned with the blow-up and global existence of nonnegative solutions to the following Cauchy problemut−Δu=vp,t>0,x∈RN,vt−Δv=a(x)uq,t>0,x∈RN,u(x,0)=u0(x),v(x,0)=v0(x),x∈RN, where the constants p,q>0p,q>0 and a(x)≩0a(x)≩0 is on the order |x|m|x|m as |x|→∞|x|→∞, m∈Rm∈R. The Fujita critical exponent is determined when m≥0m≥0, and some results of global existence of solution under some assumptions when m<0m<0 are also obtained. The results extend those in Escobedo and Herrero (1991) [9] and indicate that m affects the Fujita critical exponent.