Attractors for the nonclassical diffusion equations with fading memory

Abstract

We discuss long-time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical. The existence and regularity of global attractors in weak topological space and strong topological space are obtained, while the forcing term only belongs to H −1 (Ω) and L 2 (Ω) respectively. The results in this part are new and appear to be optimal corresponding to the forcing term.

Nonclassical diffusion with memory

“…In the recent work , rephrasing the problem in the past history framework of Dafermos , and assuming , a dissipative solution semigroup S ( t ) is obtained, acting on a suitable phase space accounting for the past values of the variable u . The semigroup is shown to possess the global attractor.…”

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…”

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.…”

“…The system has been studied in [4,23] where it is shown to generate a dissipative semigroup onĤ. Precisely, the following result holds.…”

“…Proof We limit ourselves to the validity of estimate (5.3), since the well-posedness is obtained recasting in the minimal state setting the proof performed in the memory framework, see [4,23]. Testing within a regularization scheme system (5.1) with (u, ξ) ∈ H × S, and appealing to (5.2), we get…”