Continuity and pullback attractors for a non-autonomous reaction–diffusion equation in [formula omitted

Abstract

In this paper, we study the dynamics of a non-autonomous reaction–diffusion equation in RN with the nonlinearity f satisfying the polynomial growth of arbitrary order p−1(p≥2). Firstly, we prove the existence of the pullback Dα-attractor in L2(RN). Secondly, we use a scheme to establish some new estimates, higher-order integrability of the difference of the solutions near the initial time, instead of using the usual estimates about higher regularities and higher-order integrability of solutions. Thirdly, we verify that the usual (L2(RN),L2(RN))-pullback Dα-attractor indeed can pullback attract the Dα-class in L2+δ-norm for any δ∈[0,∞) and H1-norm, and the solutions of the equation in H1(RN) are continuous with respect to initial data. Finally, we obtain the pullback Dα-attractor in Lp(RN) and H1(RN) as an application of the higher-order integrability and the continuity respectively.