Continuity of the flows and upper semicontinuity of global attractors for ps(x)-Laplacian parabolic problems

In this work we prove continuity of solutions with respect to initial conditions and exponent parameters and we prove upper semicontinuity of a family of global attractors for one-dimensional problems of the form ∂us∂t−∂∂x(|∂us∂x|ps(x)−2∂us∂x)=B(us) where B is a globally Lipschitz map, ps(⋅)→pinL∞(I)(I:=(c,d)andp>2constant) as s goes to infinity.