Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent

In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations in locally uniform spaces u(t) - Delta ut - Delta u + f (u) = g(x), x is an element of R-N. First, we prove the well-posedness of solution for the nonclassical diffusion equations with critical nonlinearity in locally uniform spaces, and then the existence of (H-lu(1)(R-N), H rho(1)(R-N))-global attractor is established. Finally, we obtain the asymptotic regularity of solutions which appears to be optimal and the existence of a bounded (in (H-lu(2)(R-N))) subset which attracts exponentially every initial H-lu(1)(R-N)- bounded set with respect to the H-lu(1)(R-N)- norm.