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On the hyperbolic relaxation of the Cahn–Hilliard equation with a mass source 有质量源的Cahn–Hilliard方程的双曲松弛
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-06-14 DOI: 10.3233/asy-231844
Dieunel Dor
{"title":"On the hyperbolic relaxation of the Cahn–Hilliard equation with a mass source","authors":"Dieunel Dor","doi":"10.3233/asy-231844","DOIUrl":"https://doi.org/10.3233/asy-231844","url":null,"abstract":"In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41988110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains 类lipschitz区域中二维多时滞Navier-Stokes方程的动力学和鲁棒性
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-05-31 DOI: 10.3233/asy-231845
Keqin Su, Xinguang Yang, A. Miranville, He Yang
{"title":"Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains","authors":"Keqin Su, Xinguang Yang, A. Miranville, He Yang","doi":"10.3233/asy-231845","DOIUrl":"https://doi.org/10.3233/asy-231845","url":null,"abstract":"This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43612584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic stability of stationary solutions for the Kirchhoff equation Kirchhoff方程平稳解的渐近稳定性
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-04-05 DOI: 10.3233/asy-231841
Min Yu, Weijia Li, Weiping Yan
{"title":"Asymptotic stability of stationary solutions for the Kirchhoff equation","authors":"Min Yu, Weijia Li, Weiping Yan","doi":"10.3233/asy-231841","DOIUrl":"https://doi.org/10.3233/asy-231841","url":null,"abstract":"This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41877431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Clausius–Mossotti formula 克劳修斯-莫索蒂公式
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-31 DOI: 10.3233/asy-231840
Mitia Duerinckx, A. Gloria
{"title":"The Clausius–Mossotti formula","authors":"Mitia Duerinckx, A. Gloria","doi":"10.3233/asy-231840","DOIUrl":"https://doi.org/10.3233/asy-231840","url":null,"abstract":"In this note, we provide a short and robust proof of the Clausius–Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and it can be applied to dilute systems in many other contexts.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48940137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Well-posedness and polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction 热传导Rayleigh梁模型传输问题的适定性和多项式能量衰减率
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-31 DOI: 10.3233/asy-231849
Mohammad Akil, Mouhammad Ghader, Z. Hajjej, Mohamad Ali Sammoury
{"title":"Well-posedness and polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction","authors":"Mohammad Akil, Mouhammad Ghader, Z. Hajjej, Mohamad Ali Sammoury","doi":"10.3233/asy-231849","DOIUrl":"https://doi.org/10.3233/asy-231849","url":null,"abstract":"In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem’s well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt–Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44822168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy 弱适定双曲边值问题的几何光学展开:掠光简并
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-14 DOI: 10.3233/asy-231838
Antoine Benoit, R. Loyer
{"title":"Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy","authors":"Antoine Benoit, R. Loyer","doi":"10.3233/asy-231838","DOIUrl":"https://doi.org/10.3233/asy-231838","url":null,"abstract":"This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l’Institut Fourier 60 (2010) 2183–2233) and (Differential Integral Equations 27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 1347–1361; SIAM J. Math. Anal. 44 (2012) 1925–1949) for the wave equation with Neumann boundary condition.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42874583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Internal layer intersecting the boundary of a domain in a singular advection–diffusion equation 奇异平流扩散方程中与区域边界相交的内层
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-13 DOI: 10.3233/asy-231836
Y. Amirat, A. Münch
{"title":"Internal layer intersecting the boundary of a domain in a singular advection–diffusion equation","authors":"Y. Amirat, A. Münch","doi":"10.3233/asy-231836","DOIUrl":"https://doi.org/10.3233/asy-231836","url":null,"abstract":"We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0, ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41877961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bifurcation from infinity and multiplicity results for an elliptic system from biology 生物学上的椭圆系统的无穷分岔和多重性结果
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-13 DOI: 10.3233/asy-231839
Chunqiu Li, Zhen Peng
{"title":"Bifurcation from infinity and multiplicity results for an elliptic system from biology","authors":"Chunqiu Li, Zhen Peng","doi":"10.3233/asy-231839","DOIUrl":"https://doi.org/10.3233/asy-231839","url":null,"abstract":"This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology − κ Δ u = λ u + f ( x , u ) − v , − Δ v = u − v , in a bounded domain Ω ⊂ R N . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45299308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Singular elliptic problem involving a Hardy potential and lower order term 涉及Hardy势和低阶项的奇异椭圆问题
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-09 DOI: 10.3233/asy-231832
A. Sbai, Y. El hadfi, Mounim El Ouardy
{"title":"Singular elliptic problem involving a Hardy potential and lower order term","authors":"A. Sbai, Y. El hadfi, Mounim El Ouardy","doi":"10.3233/asy-231832","DOIUrl":"https://doi.org/10.3233/asy-231832","url":null,"abstract":"We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in  Ω u > 0 in  Ω u = 0 on  ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω )  and  1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42479028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
D s , 2 ( R N ) versus C ( R N ) local minimizers D s，2（RN）与C（RN）局部极小值
IF 1.4 4区 数学
Asymptotic Analysis Pub Date : 2023-03-07 DOI: 10.3233/asy-231833
V. Ambrosio
{"title":"D s , 2 ( R N ) versus C ( R N ) local minimizers","authors":"V. Ambrosio","doi":"10.3233/asy-231833","DOIUrl":"https://doi.org/10.3233/asy-231833","url":null,"abstract":"Let s ∈ ( 0 , 1 ), N > 2 s and D s , 2 ( R N ) : = { u ∈ L 2 s ∗ ( R N ) : ‖ u ‖ D s , 2 ( R N ) : = ( C N , s 2 ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 ( R N ) → R of the type J ( u ) : = 1 2 ‖ u ‖ D s , 2 ( R N ) 2 − ∫ R N b ( x ) G ( u ) d x , where G ( t ) : = ∫ 0 t g ( τ ) d τ, g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 ( R N ) : u ∈ C ( R N )  with  sup x ∈ R N ( 1 + | x | N − 2 s ) | u ( x ) | < ∞ } must be a local minimizer of J in the D s , 2 ( R N )-topology.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44010650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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