On Some Model Diffusion Problems With a Nonlocal Lower Order Term

Chang, Nai-Heng; Chipot, Michel

doi:10.1142/s0252959903000153

Cited by 27 publications

(21 citation statements)

References 9 publications

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“…As said before, a kind of Lyapunov structure were available only in very special situations (see [12,14,15]). Another modification of the above equation, with the nonlocal term in another position rather than in the diffusion, was also treated in [8], with similar conclusions (see also [7] for a model that includes as particular cases those cited above).…”

Section: Introductionmentioning

confidence: 91%

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

Caraballo

Herrera-Cobos

Marín-Rubio

2015

Nonlinear Analysis: Theory, Methods & Applications

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This paper is devoted to study the asymptotic behaviour of a time-dependent parabolic equation with nonlocal diffusion and nonlinear terms with sublinear growth. Namely, we extend some previous results from the literature, obtaining existence, uniqueness, and continuity results, analyzing the stationary problem and decay of the solutions of the evolutionary problem, and finally, under more general assumptions, ensuring the existence of pullback attractors for the associated dynamical system in both L 2 and H 1 norms. Relationship among these objects are established using regularizing properties of the equation.

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Section: Introductionmentioning

confidence: 91%

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

Caraballo

Herrera-Cobos

Marín-Rubio

2015

Nonlinear Analysis: Theory, Methods & Applications

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“…In addition, by (20), these functions are non-increasing on [τ, T ]. Now, taking into account (18), one deduces that…”

Section: Proposition 1 Under the Assumptions Of Lemma 1 Ifmentioning

confidence: 99%

“…In addition, some results that establish order relationships among two stationary solutions and the long-time behaviour of the solution of the evolution problem have also been studied (cf. [23,24,[17][18][19]). …”

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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“…where b is a positive continuous function. • In the context of reaction-diffusion systems, it is also frequent to find terms of this kind; the particular case B(u(·, t), t) = b ( L, u(·, t) ), where b is a real positive continuous function and L is a continuous linear form on L 2 (Ω), has been investigated for instance by Chang and Chipot [2]. • In the context of hyperbolic equation, terms of the kind B(u(·, t), t) = b Ω |∇u| 2 , appear in the Kirchhoff equations, which arises in nonlinear vibration theory; see for instance [16].…”

Section: Introductionmentioning

confidence: 99%

Local null controllability for degenerate parabolic equations with nonlocal term

Demarque

Límaco

Viana

2018

Nonlinear Analysis: Real World Applications

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We establish a local null controllability result for following the nonlinear parabolic equation:where b(x, r) = (r)a(x) is a function with separated variables that defines an operator which degenerates at x = 0 and has a nonlocal term. Our approach relies on an application of Liusternik's inverse mapping theorem that demands the proof of a suitable Carleman estimate.2010 Mathematics Subject Classification. Primary 35K65, 93B05; Secondary 35K55.

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On Some Model Diffusion Problems With a Nonlocal Lower Order Term

Abstract: We consider a class of nonlinear parabolic problems where the lower order term is depending on a weighted integral of the solution. We address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.

Cited by 27 publications

References 9 publications

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

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

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

Local null controllability for degenerate parabolic equations with nonlocal term

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