Local existence and nonexistence for reaction-diffusion systems with coupled exponential nonlinearities

Abstract

We study the reaction-diffusion system with coupled exponential nonlinearities{∂tu=Δu+ep1u+p2vin Ω×(0,T),∂tv=Δv+eq1u+q2vin Ω×(0,T),u(x,t)=v(x,t)=0on ∂Ω×(0,T),u(x,0)=u0(x),v(x,0)=v0(x)in Ω, where T>0, pi≥0 and qi≥0(i=1,2) with (p1,p2)≠(0,0) and (q1,q2)≠(0,0). The domain Ω is RN(N≥1) or a bounded domain in RN with C2 boundary and the initial functions u0 and v0 are nonnegative and measurable. For each (p1,p2,q1,q2), we obtain integrability conditions of (u0,v0) which explicitly determine the existence/nonexistence of a local in time nonnegative classical solution. For the existence result, we can take a wider class of initial functions than previously suggested. Our analysis can be applied to other nonlinearities including superexponential ones.