UNIFORM ATTRACTORS FOR NON-AUTONOMOUS NONCLASSICAL DIFFUSION EQUATIONS ON ℝ^N

Abstract

Abstract. We prove the existence of uniform attractors Aε in the space H 1 (R N )∩L p (R N ) for the following non-autonomous nonclassical diffusion equations on R N ,The upper semicontinuity of the uniform attractors {Aε} ε∈ [0,1] at ε = 0 is also studied.

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

“…In the last two decades, many researchers have concentrated on the theory of attractors for dynamical systems. The existence and long‐time behavior of solutions to nonclassical diffusion equations have been investigated extensively in various cases, such as in the cases of autonomous (see other works 5‐11 ) and of nonautonomous (see other studies 7,12‐29 ), and even in the case with finite delay (see other works 21‐23 ). Besides, these equations with singularly oscillating external force have been also studied in Anh and Toan, 24 where they obtained the boundedness and the upper semicontinuity of uniform attractors in the case when the domain is unbounded and the nonlinearity is of Sobolev type.…”

Dynamical behavior of nonclassical diffusion equation with perturbed parameter and memory

“…This problem has been considered by some authors. In [2,5], the upper semicontinuity of uniform(pullback) attractors in L 2 (Ω) for equation (1) with subcritical nonlinearity was considered. In [4,24], the upper semicontinuity of global attractors in H 1 (R n ) for equation (1) defined in unbounded domains with subcritical nonlinearity was considered.…”

“…In order to obtain the continuity of uniform attractors for perturbation parameter ν, we need to verify the continuity for the family of processes (w.r.t ν), which is the strong continuity from the uniform bounded of ∂ t u in H 1 0 (Ω). This maybe a reason why some authors (see e.g., [4,5,26]) have to assume further that the external forcing term g(x, t) is differentiable for time term and…”