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 operatorsut+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 equationut−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).