On a class of problems involving a nonlocal operator

Corrêa, Francisco Júlio S. A.; Menezes, Silvano Dias Bezerra de; Ferreira, J.

doi:10.1016/s0096-3003(02)00740-3

Cited by 66 publications

(36 citation statements)

References 3 publications

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“…In addition, from this, (29) and (30), it can be proved analogously as in the proof of Proposition 1 that…”

Section: Proofmentioning

confidence: 62%

“…Then, after passing to the limit, bearing in mind (29) and the fact that u n τ ⇀ u τ weakly in L 2 (Ω), we deduce…”

Section: Proofmentioning

confidence: 90%

“…Now, applying Theorem 4, we deduce that there exists u 0 ∈ Φ 0 (τ, u τ ) and a subsequence of {ε n } n≥1 (relabeled the same) such that (29) holds in (τ (t, D ε0 0 , δ), t). In particular, from the last convergence in (29) at time t, we deduce that there exists n 0 ≥ 1 such that…”

Section: Proofmentioning

confidence: 91%

“…with homogeneous Dirichlet boundary conditions was studied in [12] when f was a sublinear function (see also [29] for a particular close result with exponential decay to zero). Later, we have proved an analogous result (submitted for publication) for f satisfying…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

2015

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In this paper, we consider a non-autonomous nonlocal reactiondiffusion equation with a small perturbation in the nonlocal diffusion term and the non-autonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the L 2 -norm are analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated to the autonomous nonlocal limit problem when the parameter goes to zero.Keywords nonlocal diffusion · reaction-diffusion equations without uniqueness · pullback attractors · upper semicontinuity of attractors · multivalued dynamical systems.

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“…In addition, from this, (29) and (30), it can be proved analogously as in the proof of Proposition 1 that…”

Section: Proofmentioning

confidence: 62%

“…Then, after passing to the limit, bearing in mind (29) and the fact that u n τ ⇀ u τ weakly in L 2 (Ω), we deduce…”

Section: Proofmentioning

confidence: 90%

Section: Proofmentioning

confidence: 91%

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

2015

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“…Existence of positive steady state solutions of non-local differential equations has been studied previously by many authors (e.g. [1,6,8,9,21,23,25,29]). However, the effect of the non-local terms, in isolation, on the existence of positive solutions been not been previously considered.…”

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confidence: 99%

Existence of Positive Solutions due to Non-local Interactions in a Class of Nonlinear Boundary Value Problems

Davidson¹,

Dodds²

2007

Methods and Applications of Analysis

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Abstract. We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.

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A topological approach to nonlocal elliptic partial differential equations on an annulus

Goodrich

2020

Mathematische Nachrichten

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On a class of problems involving a nonlocal operator

Cited by 66 publications

References 3 publications

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

Robustness of nonautonomous attractors for a family of nonlocal reaction–diffusion equations without uniqueness

Existence of Positive Solutions due to Non-local Interactions in a Class of Nonlinear Boundary Value Problems

A topological approach to nonlocal elliptic partial differential equations on an annulus

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