A critical exponent of Fujita type for a nonlinear reaction–diffusion system on a Riemannian manifold

In this paper, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction–diffusion system {ut−Δu=W(x)vp+S(x)in Mn×(0,∞),vt−Δv=F(x)ud+G(x)in Mn×(0,∞),u(x,0)=u0(x)in Mn,v(x,0)=v0(x)in Mn, where Mn(n≥3) is a non-compact complete Riemannian manifold, ΔΔ is the Laplace–Beltrami operator, and S(x),G(x)S(x),G(x) are non-negative Lloc1 functions. We assume that both u0(x)u0(x) and v0(x)v0(x) are non-negative, smooth and bounded functions, and constants p,d>1. When p=dp=d, there is an exponent p∗p∗ which is critical in the following sense. When p∈(1,p∗]p∈(1,p∗], the above problem has no global positive solution for any non-negative constants S(x),G(x)S(x),G(x) not identically zero. When p∈[p∗,∞)p∈[p∗,∞), the problem has a global positive solution for some S(x),G(x)>0S(x),G(x)>0 and u0(x),v0(x)≥0u0(x),v0(x)≥0.