Global attractors for multivalued semiflows with weak continuity properties

A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.Comment: to appear in Nonlinear Analysis Series A, Theory, Methods & Application

Minimality properties of set-valued processes and their pullback attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.Comment: 33 pages. A few typos correcte

Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-convex part as well as a semilinear parabolic equation with a nonlinear term involving gradients

Aronszajn-Hukuara type theorem for semilinear differential inclusions with nonlocal conditions

In this note we investigate the topological structure of the mild solution set of nonlocal Cauchy problems governed by semilinear differential inclusions in separable Banach spaces. We show that the mild solution set is a compact absolute retract (and then a continuum and $R_\delta$-set). As a particular case, the topological structure of the periodic mild solution set is deduced. An illustrating example is supplied