Continuity and pullback attractors for a non-autonomous reaction–diffusion equation in RNRN

In this paper, we study the dynamics of a non-autonomous reaction–diffusion equation in RNRN with the nonlinearity ff satisfying the polynomial growth of arbitrary order p−1(p≥2)p−1(p≥2). Firstly, we prove the existence of the pullback DαDα-attractor in L2(RN)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))(L2(RN),L2(RN))-pullback DαDα-attractor indeed can pullback attract the DαDα-class in L2+δL2+δ-norm for any δ∈[0,∞)δ∈[0,∞) and H1H1-norm, and the solutions of the equation in H1(RN)H1(RN) are continuous with respect to initial data. Finally, we obtain the pullback DαDα-attractor in Lp(RN)Lp(RN) and H1(RN)H1(RN) as an application of the higher-order integrability and the continuity respectively.