Critical exponent for parabolic system with time-weighted sources in bounded domain

This paper considers the time-weighted parabolic system ut=Δu+eαtvput=Δu+eαtvp, vt=Δv+eβtuqvt=Δv+eβtuq in bounded domain with α,β∈Rα,β∈R and p,q>0p,q>0, subject to null Dirichlet boundary condition. The critical Fujita curve is determined as (pq)c=1+max{α+βp,β+αq,0}λ1, where λ1λ1 is the first eigenvalue of the Laplacian. As an extension, it is observed for another coupled system Ut=ΔU+mU+VpUt=ΔU+mU+Vp, Vt=ΔV+nV+UqVt=ΔV+nV+Uq with pq>1pq>1 that there is the Fujita critical coefficient  max{m,n}=λ1max{m,n}=λ1, namely, any nontrivial solution blows up in finite time if and only if max{m,n}⩾λ1max{m,n}⩾λ1. The studies of critical curves for coupled systems in the current literature are all heavenly relying upon Jensenʼs inequality and the Kaplan method, for which one has to deal with complicated discussions on the exponents p, q being greater or less than 1. Differently, in the present framework, the heat semigroup is introduced to study critical curves for coupled systems, where various superlinear and sublinear cases can be treated uniformly by estimates involved. This greatly simplifies the arguments for establishing Fujita type theorems.