Attractors for the non-autonomous nonclassical diffusion equations with fading memory

Nonclassical diffusion with memory

“…However, no regularity of the attractor is given, not even for f ∈ L 2 (Ω). In fact, such a regularity when f ∈ L 2 (Ω) is implicitly proved in , which deals with the nonautonomous case.…”

Attractors for a Class of Abstract Evolution Equations with Fading Memory

“…Now we will establish the necessary asymptotic smoothness, similar to that in [17][18][19]; we also need to make some decompositions about nonlinearity, forcing term, solution, and solution semigroup. About forcing term ( ), we think about two cases: the weak solutions case ( ) ∈ −1 (Ω): for every ∈ −1 (Ω) and any > 0, there is ∈ 2 (Ω) which depends on and , such that…”

“…Then, for any given 0 = ( 0 , 0 , 0 ) ∈ H 2 , and any > 0, there exists a unique solution ( ) = ( , , ) ∈ H 2 for problem (17)- (18), in the sense that…”

A Remark on Nonclassical Diffusion Equations with Memory

“…The well-posedness and the long-term behavior of the solutions to (1.1) have been studied in [22,23] (see also [20] and the references therein for the model without memory) for a smaller class of smooth memory kernels with μ satisfying μ (s) + δμ(s) ≤ 0.…”