Nonlocal diffusion equations with dynamical boundary conditions

Berná, Pablo M.; Rossi, Julio D.

doi:10.1016/j.na.2020.111751

Cited by 11 publications

(3 citation statements)

References 23 publications

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“…The study of nonlocal problems modeled by partial differential equations has been receiving much attention recently, as the published literature confirm (see, for instance, [1,5,6,13,[16][17][18]26,27,29,30] and the references cited therein). Just to explain the interest of this type of models, let us describe two situations in which nonlocal equations are fully justified: one is related to population dynamics and the other to physical problems involving heat transfer.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of stochastic nonlocal partial differential equations

Caraballo

2021

Eur. Phys. J. Plus

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This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations, we then construct the dynamical systems to the original problems via the properties of conjugation. Next, in the case of multiplicative noise, we establish the existence of the random attractor when it absorbs every bounded deterministic set. Particularly, it is shown the pullback random attractor, which is also forward attracting, becomes a singleton when the external forcing term vanishes at zero. Eventually, in the case of additive noise, two approaches are applied to prove the existence of pullback random attractors with the help of energy estimations. Actually, these two attractors turn out to be the same one.

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Section: Introductionmentioning

confidence: 99%

“…Therefore, this paper is devoted to study the long-time behavior of the following two stochastic nonlocal partial differential equations with multiplicative and additive noise, respectively, (5) where, as we mentioned, W (t) is a standard Brownian motion, and • denotes the Stratonovich sense in the stochastic term.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of stochastic nonlocal partial differential equations

Caraballo

2021

Eur. Phys. J. Plus

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“…For example, the model in [26] focused on the role of nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. The authors in [4] studied nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Non-autonomous nonlocal partial differential equations with delay and memory

Zhang

Caraballo

2021

Journal of Differential Equations

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The paper addresses a kind of non-autonomous nonlocal parabolic equations when the external force contains hereditary characteristics involving bounded and unbounded delays. First, well-posedness of the problem is analyzed by the Galerkin method and energy estimations in the phase space C ρ (X). Moreover, some results related to strong solutions are proved under suitable assumptions. The existence of stationary solutions is then established by a corollary of the Brower fixed point theorem. By constructing appropriate Lyapunov functionals in terms of the characteristic delay terms, a deep analysis on stability and attractiveness of the stationary solutions is established. Furthermore, the existence of pullback attractors in L 2 (Ω), with bounded and unbounded delays, is shown. We emphasize that, to prove the existence of pullback attractors in the unbounded delay case, a new phase space, E γ , has to be constructed.

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