Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

Abstract

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.