Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations

In this paper, we introduce the concept of norm-to-weak continuous process in a Banach space, and obtain the existence of pullback attractors for this kind of process. Then we give a new method for proving the existence of the pullback attractors. As an application, we obtain the existence of pullback attractors for nonautonomous reaction-diffusion equation in H01 with exponential growth of the external force.