EVOLUTION EQUATIONS ON TIME-DEPENDENT LEBESGUE SPACES WITH VARIABLE EXPONENTS

We extend the results in Kloeden-Simsen [CPAA 2014] to p(x, t)-Laplacian problems on time-dependent Lebesgue spaces with variable expo-nents. We study the equation partial differential uA (t) div (DA(t, x)|vuA(t)|p(x,t)-2vuA(t)) + |uA(t)|p(x,t)-2uA(t) partial differential t = B(t, uA(t)) on a bounded smooth domain & omega; in R , n> 1, with a homogeneous Neumann boundary condition, where the exponent p(& BULL;) E C(& omega; over bar x [& tau;, T], R+) satisfies min p(x, t) > 2, and & lambda; E [0, oo) is a parameter.We establish the existence and upper semicontinuity of pullback attractors for this equation under the assumption, amongst others, that B is globally Lipschitz in its second variable and DA E L & INFIN;([& tau;, T] x & omega;, R+) is bounded from above and below, monotonically nonincreasing in time and continuous in the parameter & lambda;.