Global existence, blow-up and asymptotic behavior of solutions for a class of p(x)-Choquard diffusion equations in [formula omitted

Abstract

In this paper, we investigate the local and global existence, asymptotic behavior, and blow-up of solutions to the Cauchy problem for Choquard-type equations involving the p(x)-Laplacian operator. As a particular case, we study the following initial value problem{ut−Δp(x)u+V(x)|u|p(x)−2u=(∫RN|u|q(y)q(y)|x−y|α(x,y)dy)|u|q(x)−2uinRN×(0,+∞),u(x,0)=u0(x),inRN, where p,q,V:RN→R and α:RN×RN→R are continuous functions that satisfy some conditions which will be stated later on, and u0:RN→R is the initial function. Under some appropriate conditions, we prove the local and global existence of solutions for the above Cauchy problem by employing the abstract Galerkin approximation. Moreover, the blow-up of solutions and large-time behavior are also investigated.