PDE and ODE limit problems for p(x)p(x)-Laplacian parabolic equations

In this work we prove continuity of solutions with respect to initial conditions and parameters and we prove upper semicontinuity of a family of global attractors for problems of the formut−div(Dλ|∇uλ|p(x)−2∇uλ)=B(uλ)ut−div(Dλ|∇uλ|p(x)−2∇uλ)=B(uλ) in a bounded smooth domain Ω   in RNRN.