On pullback attractors in H10 for nonautonomous reaction-diffusion equations

Using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set together with some new estimates of solutions we prove existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction-diffusion system in H10 in which the right hand side satisfies only a certain integrability condition. In particular, we generalize a result obtained recently in [10] where an at most exponential growth of the right hand side has been assumed for times going to both plus and minus infinity