Long-time behavior for a nonlinear parabolic problem with variable exponents

Niu, Weisheng

doi:10.1016/j.jmaa.2012.03.039

Cited by 15 publications

(9 citation statements)

References 25 publications

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“…In [37], Yang et al obtained the existence of global attractors for (1.7). Indeed, the existence of global attractors is an important asymptotic property of solutions for parabolic equations which have been studied extensively by many researchers, see for example [6,10,23,24,38].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this paper, inspired by the above-mentioned papers, we discuss the existence of global attractors and fractal dimension of solutions for problem (1.1) involving the fractional p-Laplacian and nonlocal diffusion coefficient. Motivated by [23,37], we first study the existence and uniqueness of solutions by using the Galerkin method. Then we give the existence of global attractors in proper spaces and obtain the fractal dimension of the global attractors.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Definition 2.1 (see [23,24]) Let {S(t)} t≥0 be a semigroup on Banach space X. A subset A ⊂ X is called a global attractor for the semigroup if A is compact in X and satisfies the following properties:…”

Section: Preliminariesmentioning

confidence: 99%

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Long-time behavior of solutions for a fractional diffusion problem

Xiang

2021

Bound Value Probl

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This paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: $$ u_{t}+M\bigl([u]_{s,p}^{p}\bigr) (-\Delta )_{p}^{s}u+f(x,u)=g(x)\quad \text{in } \Omega \times (0, \infty ), $$ u t + M ( [ u ] s , p p ) ( − Δ ) p s u + f ( x , u ) = g ( x ) in Ω × ( 0 , ∞ ) , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N is a bounded domain with Lipschitz boundary, $N>ps$ N > p s , $0< s<1<p$ 0 < s < 1 < p , $M:[0,\infty )\rightarrow [0,\infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) is a nondecreasing continuous function, $[u]_{s,p}$ [ u ] s , p is the Gagliardo seminorm of u, $f:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ f : Ω × R → R and $g\in L^{2}(\Omega )$ g ∈ L 2 ( Ω ) . With general assumptions on f and g, we prove the existence of global attractors in proper spaces. Then, we show that the fractal dimensional of global attractors is infinite provided some conditions are satisfied.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Long-time behavior of solutions for a fractional diffusion problem

Xiang

2021

Bound Value Probl

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“…When u 0 (x) ∈ L 2 ( ), the existence and localization of weak solutions were considered in [7]. At the same time, the intrinsic Harnack inequalities, the long-time behavior and the Hölder regularity of weak solutions to equation (1.4) were studied in the literature [11,14,16].…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of a nonlinear parabolic equation with double variable exponents

2021

Anal.Math.Phys.

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The well-posedness problem of a nonlinear parabolic equation with double variable exponents is studied in this paper. This kind of nonlinear parabolic equation includes the non-Newtonian fluids equation, the polytropic filtration equation and the so-called electro-rheological fluid equation. One of the important characteristics is that there is a diffusion coefficient a(x, t) in the equation. Unlike the usual assumption a(x, t) > a > 0, the paper only assumes a(x, t) ≥ 0. If a(x, t)| x∈∂ = 0, by choosing a(x, t) as a test function, the stability result of positive solutions can be established without the boundary value condition.

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“…We would like to suggest that the first paper where parabolic equations with variable growth exponent are considered is Acerbi et al [3]; some other interesting works are listed as [4][5][6][7][8][9][10][11] in our references. If ( ) = 0, ∈ Ω, then the equation is degenerate on the boundary; the author had shown that, in [12], besides the initial value condition (2), only a partial boundary value condition ( , ) = 0, ( , ) ∈ Σ × (0, ) ,…”

Section: Introductionmentioning

confidence: 99%

Diffusion Convection Equation with Variable Nonlinearities

Zhan¹

2017

Journal of Function Spaces

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The paper studies diffusion convection equation with variable nonlinearities and degeneracy on the boundary. Unlike the usual Dirichlet boundary value, only a partial boundary value condition is imposed. If there are some restrictions in the diffusion coefficient, the stability of the weak solution based on the partial boundary value condition is obtained. In general, we may obtain a local stability of the weak solutions without any boundary value condition.

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Long-time behavior for a nonlinear parabolic problem with variable exponents

Cited by 15 publications

References 25 publications

Long-time behavior of solutions for a fractional diffusion problem

Long-time behavior of solutions for a fractional diffusion problem

Positive solutions of a nonlinear parabolic equation with double variable exponents

Diffusion Convection Equation with Variable Nonlinearities

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