Nonclassical diffusion equations on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math>with singularly oscillating external forces

Anh, Cung The; Toan, Nguyen Duong

doi:10.1016/j.aml.2014.06.008

Cited by 25 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the last two decades, many researchers have concentrated on the theory of attractors for dynamical systems. The existence and long-time behavior of solutions to nonclassical diffusion equations have been investigated extensively in various cases, such as in the cases of autonomous (see other works [5][6][7][8][9][10][11] ) and of nonautonomous (see other studies 7,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], and even in the case with finite delay (see other works [21][22][23] ). Besides, these equations with singularly oscillating external force have been also studied in Anh and Toan, 24 where they obtained the boundedness and the upper semicontinuity of uniform attractors in the case when the domain is unbounded and the nonlinearity is of Sobolev type.…”

Section: Introductionmentioning

confidence: 99%

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

Nguyen

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we consider a class of nonclassical diffusion equations on R N with hereditary memory, in presence of singularly oscillating external forces depending on a positive parameter and a new class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. Under a general assumption on the memory kernel and for a very large class of nonlinearities, we prove the existence of uniform attractors in H 1 (R N) × L 2 (R + , H 1 (R N)). The uniform boundedness (w.r.t.) and the convergence of uniform attractors  as tends to 0 are also studied. Our results extend and improve some results in Anh and Toan.

show abstract

Section: Introductionmentioning

confidence: 99%

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

Nguyen

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The asymptotic behavior of solutions of this equation has received considerably less attention in the literature under the assumption that the nonlinearity satisfies arbitrary polynomial growth condition. In the ordinary case for some recent results on this equations the reader can refer to Sun et al [24] and Anh and Toan [25,26]. Hereafter in [23] the authors mention that these are some mistakes in the coauthor's earlier paper [24].…”

Section: Introductionmentioning

confidence: 99%

Uniform Attractors for Nonclassical Diffusion Equations with Memory

Xie

Zeng

2016

Journal of Function Spaces

View full text Add to dashboard Cite

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearityfsatisfies the polynomial growth of arbitrary order and the time-dependent forcing termgis only translation-bounded inLloc2(R;L2(Ω)).

show abstract

“…Introduction. In this paper, we consider the following 2D Newton-Boussinesq equation    ∂ t ξ + u∂ x ξ + v∂ y ξ = ∆ξ − Ra Pr ∂ x θ + f 0 (x, y, t) + ε −ρ f 1 ( x ε , y ε , t), ∆Ψ = ξ, u = Ψ y , v = −Ψ x , ∂ t θ + u∂ x θ + v∂ y θ = 1 Pr ∆θ + g 0 (x, y, t)…”

mentioning

confidence: 99%

“…The Newton-Boussinesq equation describes the Bénard flow. There are some works concerning problem (1). For example, Guo studied the existence and uniqueness of weak solutions of two-dimensional Newton-Boussinesq equation by spectral method and Galerkin method in [15] and [14], respectively.…”

mentioning

confidence: 99%

“…Stability of attractors for a dynamical system with some oscillating external forces is also important in natural phenomenon. Indeed, this issue has been considered by some mathematicians and engineers, see [1,3,4,5,7,8,9,10,11,19,27,28,29,30,31,33]. However, as far as we know, there is no paper dealing with the non-autonomous Newton-Boussinesq equation with rapidly oscillating terms.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor

Song¹,

Wu²

2022

EECT

View full text Add to dashboard Cite

<p style='text-indent:20px;'>We consider a non-autonomous two-dimensional Newton-Boussinesq equation with singularly oscillating external forces depending on a small parameter <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>. We prove the existence of the uniform attractor <inline-formula><tex-math id="M2">\begin{document}$ A^\varepsilon $\end{document}</tex-math></inline-formula> when the Prandtl number <inline-formula><tex-math id="M3">\begin{document}$ P_r>1 $\end{document}</tex-math></inline-formula>. Furthermore, under suitable translation-compactness and divergence type condition assumptions on the external forces, we obtain the uniform (with respect to <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>) boundedness of the related uniform attractors <inline-formula><tex-math id="M5">\begin{document}$ A^\varepsilon $\end{document}</tex-math></inline-formula> as well as the convergence of the attractor <inline-formula><tex-math id="M6">\begin{document}$ A^\varepsilon $\end{document}</tex-math></inline-formula> to the attractor <inline-formula><tex-math id="M7">\begin{document}$ A^0 $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon\rightarrow 0^+ $\end{document}</tex-math></inline-formula>.</p>

show abstract

Nonclassical diffusion equations onRNwith singularly oscillating external forces

Cited by 25 publications

References 10 publications

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

Uniform attractors of nonclassical diffusion equations on ℝN with memory and singularly oscillating external forces

Uniform Attractors for Nonclassical Diffusion Equations with Memory

Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor

Contact Info

Product

Resources

About