Pullback attractors for non-autonomous evolution equations with spatially variable exponents

Abstract

Dissipative problems in electrorheological uids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation μλ/@t(t)-div (Dλ(t) μΔ(t)p(x)-2μλ(t))+μλ(t)p(x) -2μλ(t) = B(t; μλ(t))on a bounded smooth domain ω in Rn, n ≤ 1, with a homogeneous Neumann boundary condition, where the exponent p(̇) ε C(ω,R+) satisfying p := min p(x) > 2, and λ epsi; [0;∞) is a parameter. The existence and upper semicontinuity of pullback attractors are established for this equation under the assumptions, amongst others, that B is globally Lipschitz in its second variable and Dλ ε L∞([τ, T] ωR+) is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter λ. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.