Bi-spatial Pullback Attractors of Fractional Nonclassical Diffusion Equations on Unbounded Domains with (p, q)-Growth Nonlinearities

Wang, Renhai; Li, Yangrong; Wang, Bixiang

doi:10.1007/s00245-019-09650-6

Cited by 24 publications

(11 citation statements)

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“…Note that the last integral in ( 54) is indeed well defined due to (48) and (14). It yields from ( 14) and e z( −t ) v −t ∈ D( − t, −t ) with D ∈ that as t → +∞, which shows that there exists T ∶= T(𝛼, 𝜎, 𝜏, 𝜔, D) > 2 such that for all t ≥ T, Hence, the desired results ( 49)-( 51) follow from ( 54)- (55). ◻…”

Section: Necessary Uniform Estimatesmentioning

confidence: 71%

Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

Wang

2020

Banach J. Math. Anal.

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This article is concerned with the random dynamics of a wide class of nonautonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space ℝ N . We first establish the wellposedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders p, q ≥ 2 . We then prove that the equation has a unique bi-spatial pullback random attractor that is measurable, compact in L 2 (ℝ N ) ∩ L p (ℝ N ) ∩ L q (ℝ N ) and attracts all random subsets of L 2 (ℝ N ) under the topology of L 2 (ℝ N ) ∩ L p (ℝ N ) ∩ L q (ℝ N ) . In addition, we establish the upper semicontinuity of these attractors in L 2 (ℝ N ) ∩ L p (ℝ N ) ∩ L q (ℝ N ) when the density of noise shrinks to zero. The idea of uniform tail estimates and the method of asymptotic a priori estimates are applied to prove the pullback asymptotic compactness of the solutions in L 2 (ℝ N ) ∩ L p (ℝ N ) ∩ L q (ℝ N ) to overcome the non-compactness of Sobolev embeddings on ℝ N as well as the almost sure nondifferentiability of the sample paths of the Wiener process.

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Section: Necessary Uniform Estimatesmentioning

confidence: 71%

Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

Wang

2020

Banach J. Math. Anal.

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“…In this special case, the nonclassical diffusion equation has been studied in [2,[4][5][6][7][8]11,14,[19][20][21] with various directions and motivations. Until now, the results on fractional pseudo-parabolic equations are limited and can be mentioned in just a few papers, for example, [7,22,28,29]. Since the fractional operator (−Δ) α appearing in the main equation is nonlocal and can be regarded as the infinitesimal generator of Levy stable diffusion processes, many scientists believed that it described some physical phenomena more exactly than integral differential equations.…”

Section: N H Tuan and T Caraballomentioning

confidence: 99%

On initial and terminal value problems for fractional nonclassical diffusion equations

Tuan

Caraballo

2020

Proc. Amer. Math. Soc.

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In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case 0 < α ≤ 1 and show the ill-posedness in the sense of Hadamard in the case α > 1. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in L q norm is first established.

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“…The deterministic case of NDEs have been studied in several papers [12,14,15,19,23] and references therein. The stochastic case of NDEs can be found in few papers, for instance, B. Wang et al [40,41,42], L. Bai et al [5]. It is wellknown that stochastic NDEs and stochastic partial differential equations (SPDE) play important and fundamental roles in modeling phenomena in physics, chemistry, economics, social sciences, finance, and engineering.…”

Section: Introductionmentioning

confidence: 99%

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

et al. 2021

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This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

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Bi-spatial Pullback Attractors of Fractional Nonclassical Diffusion Equations on Unbounded Domains with (p, q)-Growth Nonlinearities

Cited by 24 publications

References 50 publications

Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

Random dynamics of non-autonomous fractional stochastic p-Laplacian equations on $${\mathbb {R}}^N$$

On initial and terminal value problems for fractional nonclassical diffusion equations

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

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