Existence and upper semicontinuity of global attractors for a $p$-Laplacian inclusion

Abstract

In this work we study the asymptotic behavior of a p-Laplacian inclusion of the formis the set of all nonempty, bounded, closed, convex subsets of R. We prove the existence of a global attractor in L 2 (Ω) for each positive finite diffusion coefficient and we show that the family of attractors behaves upper semicontinuously on positive finite diffusion parameters.

Continuity of the flows and upper semicontinuity of global attractors forps(x)-Laplacian parabolic problems

“…The reason for this is that in [10] was proved that cl H (D(A H )) = H for 2 < p(x) ≤ 3 − δ, δ > 0. But, the authors in [29] answer the question raised in [10] and now we know that cl H (D(A H )) = H for p − > 2. So, we have that for each s ∈ N such that p − s > 2 there exists a global attractor A s in H for the problem (1).…”

“…We also observe (see [29]) that the operator −∆ p (x) is the subdifferential ∂ϕ p(x) of the convex, proper and lower semicontinuous map ϕ p(x) : L 2 (Ω) → R ∪ {+∞} given by…”

Global attractors for $p$-Laplacian differential inclusions in unbounded domains

“…In this paper we will extend the results in [51] by considering unbounded domains first in the case where f is Lipschitz in the multivalued sense in both autonomous and nonautonomous cases, and after that a more general situation where f just satisfies a suitable growth condition.…”

“…The authors in [51] considered problem (1) on bounded domains when f is Lipschitz in the multivalued sense and a(x) ≡ 1 (see [31,32] for differential equations and inclusions generated by pseudo-monotone operators in the case where f ≡ 0).…”

Attractors of multivalued semi-flows generated by solutions of optimal control problems

“…The application of this theory to various classes of dissipative evolution systems without uniqueness can be found in a great number of papers in both the autonomous (multivalued semiflows) and nonautonomous (multivalued semiprocesses) cases (see e.g. [3], [4], [10], [11], [18], [21], [22], [23], [24], [26], [29], [32], [33], [35], [37], [38], [39], [40], [41] among many others).…”