Evolution inclusions governed by subdifferentials in reflexive Banach spaces

Abstract

The existence, uniqueness and regularity of strong solutions for Cauchy problem and periodic problem are studied for the evolution equation:∂ϕ is the so-called subdifferential operator from a real Banach space V into its dual V * . The study in the Hilbert space setting (V = V * = H : Hilbert space) is already developed in detail so far. However, the study here is done in the V -V * setting which is not yet fully pursued. Our method of proof relies on approximation arguments in a Hilbert space H . To assure this procedure, it is assumed that the embeddings V ⊂ H ⊂ V * are both dense and continuous.

On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian

“…Finally, some blow-up results were also proved in [13] for (1.3) with h (τ ) ∼ − |τ | q−2 τ , as |τ | → ∞, emphasizing the same critical blow-up exponent q = p = 2 as for the corresponding parabolic equation associated with the classical Laplace operator −∆. We extend our work of [13] to prove the local in time existence of solutions to parabolic equations with degenerate fractional diffusion and more singular kernels using an approach based on [3,4] and also developed further in [2]. Although our general scheme follows closely that of [2,3,4], many of the key lemmas used in the case of the classical p-Laplace operator cannot be adapted or exploited in their classical form to deal with the fractional p-Laplacian (−∆) s p for s ∈ (0, 1) and p ∈ (1, ∞).…”

“…We extend our work of [13] to prove the local in time existence of solutions to parabolic equations with degenerate fractional diffusion and more singular kernels using an approach based on [3,4] and also developed further in [2]. Although our general scheme follows closely that of [2,3,4], many of the key lemmas used in the case of the classical p-Laplace operator cannot be adapted or exploited in their classical form to deal with the fractional p-Laplacian (−∆) s p for s ∈ (0, 1) and p ∈ (1, ∞). Hence, we develop some new techniques including some new functional inequalities allowing us to extend the results of [2] in the present setting.…”

“…There is vast literature on degenerate parabolic equations involving the classical diffusion operator −∆ p . We refer the reader to the following list [2,3,4,6,18] (and references contained therein) which is not meant to be exhaustive.…”

Evolution Mixed Variational Inclusions with Optimal Control

“…In this last section, we state an optimal control problem, governed by primal and dual evolution mixed variational systems (M) and (M * ), performing the state existence analysis on the basis of (Akagi, G. &Ôtani, M., 2004) work on evolution inclusions. The corresponding optimal control existence is given in accordance with (Migórski, S., 2001) study on optimal control of evolution hemivariational inequalities.…”

“…The solvability analysis is performed by adapting the study of (Akagi, G. &Ôtani, M., 2004), to our primal and dual evolution mixed maximal monotone subdifferential systems. The optimality analysis is given on the basis of Migórski's work on optimal control of evolution hemivariational inequalities (Migórski, S., 2001).…”