PULLBACK ATTRACTORS FOR NON-AUTONOMOUS EVOLUTION EQUATIONS WITH SPATIALLY VARIABLE EXPONENTS

Dissipative problems in electrorheological fluids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation partial derivative u lambda/partial derivative t (t) - div (D-lambda (t)vertical bar del u(lambda)(t)vertical bar(p(x)-2)del u(lambda)(t)) + vertical bar u(lambda)(t)vertical bar(p(x)-2)u(lambda)(t) - B(t, u(lambda)(t)) on a bounded smooth domain Omega in R-n, n >= 1, with a homogeneous Neumann boundary condition, where the exponent p(.) is an element of C((Omega) over bar, R+) satisfying p(-) := min p(x) > 2, and lambda is an element of [0, infinity) 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 D-lambda is an element of L-infinity ([tau, T] x Omega, R+) is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter lambda. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.