On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

AbstractIn this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation ut−Δmu+a(x)|u|αu=f0(u)∫ΩK(y)|u(y,t)|βdy+g(x) in Ω×(0,∞) with the initial data u(x,0)=u0(x)∈Lq, q⩾1, and zero boundary condition in ∂Ω. Two cases for a(x)⩾a0>0 and a(x)⩾0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p⩾max{1,q}. The attractor A is in fact a bounded set in W01,m∩L∞ if a(x)⩾a0>0 in Ω, and A is bounded in W01,m∩Lp if a(x)⩾0 in Ω