Uniform Attractors for Non-autonomous Nonclassical Diffusion Equations on R

where ε ∈ [0, 1], the nonlinearity f and the external force g satisfy some certain conditions specified later. This equation is known as the nonclassical diffusion equation when ε > 0, and the reaction-diffusion equation when ε = 0. Nonclassical diffusion equation arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanic, and heat conduction (see, e.g., [1, 7, 13, 14]). The long-time behavior of solutions to problem (1.1) has been studied extensively in recent years, for both autonomous case [10, 11, 16, 18] and non-autonomous case [2, 3, 11]. However, to the best of our knowledge, most existing results related to the problem are valid in bounded domains, except the recent work [3] where the existence of pullback attractors of the problem (1.1) on R was proved. In this paper we will study the existence and upper semicontinuity of uniform attractors of a family of processes associated to problem (1.1) in the case of unbounded domains, the nonlinearity of polynomial type, and the external force g depending on time t. To