Attractors of multivalued semiflows generated by differential inclusions and their approximations. (English) Zbl 1014.35010

Kapustian, Alexei V.; Valero, José

Abstr. Appl. Anal. 5, No. 1, 33-46 (2000).

Let \(H\) be a real separable Hilbert space, \(\varphi:H \to(-\infty, +\infty]\) a proper convex lower semicontinuous function and let \(\partial \varphi:D (\partial\varphi) \subset H\to 2^H\) be its subdifferential. The authors consider the following differential inclusion \[ {dy\over dt} \in-\partial \varphi(y)+ F(y),\quad t\in[0,T] \]
\[ y(0)= y_0\in H, \] where \(F:H\to 2^H\) is a multivalued mapping. They study the existence of the global compact attractor of a multivalued semiflow for the above problem, extending the result of V. S. Mel’nik and J. Valero [Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 2001, No. 10, 7-12 (2001; Zbl 0983.35152)]. And also, they study the continuity and upper semicontinuity of the attractors for approximating and perturbed inclusions. The result is very nice.

Reviewer: Jong-Kyu Kim (Masan Kyungnam)

MSC:

35B41	Attractors
35B40	Asymptotic behavior of solutions to PDEs
35K85	Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
47J35	Nonlinear evolution equations
35R70	PDEs with multivalued right-hand sides
35K90	Abstract parabolic equations

Keywords:

global compact attractor; upper semicontinuity; approximating and perturbed inclusions; continuity

Citations:

Zbl 0983.35152