Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms

Caraballo, Tomás; Herrera-Cobos, Marta; Marín-Rubio, Pedro

doi:10.1016/j.na.2014.07.011

Cited by 33 publications

(30 citation statements)

References 22 publications

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“…Analogously as in [8,Remark 23 (ii)], we can extend the above result to new larger universes and obtaining new attractors.…”

Section: Abstract Results On the Theory Of Pullback Attractorssupporting

confidence: 62%

“…Remark 5.9. (i) In the context of [8], for f sublinear, the global attractor also satisfies the estimate (5.12) in L ∞ (Ω), provided that f (s)s ≤ κ − α 2 |s| p for all s ∈ R for some p ≥ 1.…”

Section: Pullback Attractors In Hmentioning

confidence: 97%

“…In [36], Menezes proves the existence and uniqueness of weak solution to (1.3) by fixed point arguments, considering that the function f is globally Lipschitz. Later, the analogous existence and uniqueness result is proved in [8] without any assumptions of smoothness on the boundary of Ω and the nonlinearity f (u) is just continuous, sublinear, and fulfills a monotonocity condition.…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 94%

“…From (1.5), we deduce that there exists β > 0 such that |f (s)| ≤ β(|s| p−1 + 1) ∀s ∈ R. (1.6) Under these assumptions, in addition to proving the existence of weak solutions, we also show the existence of pullback attractors in the L 2 -norm for the dynamical system associated to (1.3). While in the sublinear framework we also needed to impose the usual assumption lim sup |s|→∞ f (s)/s < mλ 1 (see (17) in [8]), here condition (1.5) will be enough.…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

Caraballo¹,

Herrera-Cobos²,

Marín-Rubio³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, the existence and uniqueness of weak and strong solutions for a non-autonomous nonlocal reaction-diffusion equation is proved. Next, the existence of minimal pullback attractors in the L 2 -norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition, and some relationships between them are established. Finally, we prove the existence of minimal pullback attractors in the H 1 -norm and study relationships among these new families and those given previously in the L 2 -context. The results are also new in the autonomous framework in order to ensure the existence of global compact attractors, as a particular case. 2010 MSC: 35B41, 35B65, 35K57, 35Q92, 37L30. Keywords and phrases: Non-autonomous nonlocal reaction-diffusion equations; pullback attractors; asymptotic compactness; regularity of attractors.2 T. Caraballo, M. Herrera-Cobos and P. Marín-Rubio a ∈ C(R; (0, ∞)) and there exist positive constants m, M such that 0 < m ≤ a(s) ≤ M ∀s ∈ R.(1.2) Furthermore, l is a continuous linear form from L 2 (Ω) into R. Namely,The assumptions made on the function a are sufficient in order to avoid that the solutions exist only in finite-time intervals (see [32] for more details). They do not make any additional assumptions on the function a because it depends on the type of problem intended to model. For example, in population dynamics, the monotonicity of the function must be adapted to the behaviour of the species we want to model (see [16]). To prove the uniqueness, they suppose that the function a is globally Lipschitz due to the presence of the nonlocal term. They also study the asymptotic behaviour of the solutions when f ∈ V and under additional assumptions. Later, in [17], Chipot and Molinet generalize the results obtained in [16], dealing with a continuum of steady states using dynamical systems. Many authors have been interested in analyzing some variants of problem (1.1). In [1,2], Andami Ovono and Rougirel study a problem in which the nonlocal operator does not act in the whole domain. In these two papers, the existence of radial solutions, bifurcation analysis, and their stability are analyzed. In [19], Chipot and Siegwart study the asymptotic behaviour of the solutions to problems with nonlocal diffusion and mixed boundary conditions. In [11], Chipot and Chang are also interested in the asymptotic behaviour of the solutions to nonlocal problems with two nonlocal terms and mixed boundary conditions. In particular, they prove results which establish relationships between the solution of the evolution problem and stationary solutions. These results are similar to those given in a simpler framework (see [16, Theorem 4.1] for more details). In [20], Chipot et al. consider a problem in which the nonlocal term depends on a Dirichlet integral. In this particular case, they are able to find a Lyapunov structure, which is used to study the asymptotic behaviour of the solutions.Despite all the cited advances in the case of f independent of the solution, the genera...

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“…Analogously as in [8,Remark 23 (ii)], we can extend the above result to new larger universes and obtaining new attractors.…”

Section: Abstract Results On the Theory Of Pullback Attractorssupporting

confidence: 62%

Section: Pullback Attractors In Hmentioning

confidence: 97%

Section: Introduction and Statement Of The Problemmentioning

confidence: 94%

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

See 2 more Smart Citations

Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

Caraballo¹,

Herrera-Cobos²,

Marín-Rubio³

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…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%

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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A non-local non-autonomous diffusion problem: linear and sublinear cases

Figueiredo-Sousa

Suárez

2017

Z. Angew. Math. Phys.

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Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms

Cited by 33 publications

References 22 publications

Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

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

A non-local non-autonomous diffusion problem: linear and sublinear cases

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