Pullback Attractors for Nonautonomous Reaction–Diffusion Equations With the Driving Delay Term in ℝ N

ABSTRACT In this paper, we mainly investigate the asymptotic behavior of nonautonomous reaction–diffusion equation with the driving delay term in whole space. A new method (or technique) is introduced for verifying theCL 2 ( ℝ N ) , CH 1 ( ℝ N )$$ \left({C}_{L^2\left({\mathbb{R}}^N\right)},{C}_{H^1\left({\mathbb{R}}^N\right)}\right) $$ ‐pullback‐asymptotic compactness of the family of processes (see Theorem 2). As an application, theCL 2 ( ℝ N ) , CH 1 ( ℝ N )$$ \left({C}_{L^2\left({\mathbb{R}}^N\right)},{C}_{H^1\left({\mathbb{R}}^N\right)}\right) $$ ‐pullback‐attractor is obtained. In particular, the nonlinearityf $$ f $$ satisfies the polynomial growth of arbitrary order, and the delay termg ( t , u t ) $$ g\left(t,{u}_t\right) $$ may be driven by a function with very weak assumptions, namely, just measurability, which deepens some previous results.