Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators

Abstract

In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operators ut + A(t)u = f (t, u), t > τ ; u(τ ) = u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t, x) with respect to its spatial variable x and estimates for ut in stronger spaces (Xα). We then apply those results to a nonautonomous reaction-diffusion equation ut − div(a(t, x)∇u) + u = f (t, u) with Neumann boundary condition and time-dependent diffusion. From the regularity of ut we derive the existence of classical solutions and from the estimates for ut we prove that the variation of the solution u is bounded in the long-time dynamics. We also prove the existence of pullback attractor, as well as the existence of a compact set that contains the long-time dynamics of the derivatives ut , without requiring any assumption concerning monotonicity or decay in time of a(t, x).