Uniform attractors for non-autonomous image-Laplacian equations Original Research Article

In this paper, by the Galerkin method, we give the existence of solutions for the non-autonomous pp-Laplacian equation View the MathML sourceut−div(|∇u|p−2∇u)+f(u)=g(t). After that, we explore the asymptotic behavior of the equation. The existence and the structures of the (L2(Ω),L2(Ω))(L2(Ω),L2(Ω))-uniform attractor and the View the MathML source(L2(Ω),Lq(Ω)∩W01,p(Ω))-uniform attractor are proved under the conditions below: the nonlinear term ff is supposed to satisfy the polynomial condition of arbitrary order c1|u|q−k≤f(u)u≤c2|u|q+kc1|u|q−k≤f(u)u≤c2|u|q+k and f′(u)≥−lf′(u)≥−l, where q≥2q≥2 is arbitrary; and the external force g(t)g(t) in View the MathML sourceLloc2(R,Ls(Ω)),s≥2, is translation bounded and uniformly bounded in Ls(Ω)Ls(Ω) with respect to t∈Rt∈R.