A. K. Nandakumaran, A. Sufian, Renjith Thazhathethil
In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain.
{"title":"Homogenization of elliptic PDE with L 1 source term in domains with boundary having very general oscillations","authors":"A. K. Nandakumaran, A. Sufian, Renjith Thazhathethil","doi":"10.3233/asy-221808","DOIUrl":"https://doi.org/10.3233/asy-221808","url":null,"abstract":"In the present article, we study the homogenization of a second-order elliptic PDE with oscillating coefficients in two different domains, namely a standard rectangular domain with very general oscillations and a circular type oscillating domain. Further, we consider the source term in L 1 and hence the solutions are interpreted as renormalized solutions. In the first domain, oscillations are in horizontal directions, while that of the second one is in the angular direction. To take into account the type of oscillations, we have used two different types of unfolding operators and have studied the asymptotic behavior of the renormalized solution of a second-order linear elliptic PDE with a source term in L 1 . In fact, we begin our study in oscillatory circular domain with oscillating coefficients and L 2 data which is also new in the literature. We also prove relevant strong convergence (corrector) results. We present the complete details in the context of circular domains, and sketch the proof in other domain.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45903720","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}
We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.
{"title":"A perturbed fractional p-Kirchhoff problem with critical nonlinearity","authors":"Luigi Appolloni, A. Fiscella, S. Secchi","doi":"10.3233/asy-221809","DOIUrl":"https://doi.org/10.3233/asy-221809","url":null,"abstract":"We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44066865","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}
This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.
{"title":"Topological asymptotic expansion for the full Navier–Stokes equations","authors":"M. Hassine, Sana Chaouch","doi":"10.3233/asy-221807","DOIUrl":"https://doi.org/10.3233/asy-221807","url":null,"abstract":"This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45544525","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}
In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.
{"title":"Semi-classical states for the Choquard equations with doubly critical exponents: Existence, multiplicity and concentration","authors":"Yujian Su, Zhisu Liu","doi":"10.3233/asy-221799","DOIUrl":"https://doi.org/10.3233/asy-221799","url":null,"abstract":"In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43952964","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}
This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α → 0. The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor.
{"title":"On a Cahn–Hilliard–Oono model for image segmentation","authors":"Lu Li","doi":"10.3233/asy-221801","DOIUrl":"https://doi.org/10.3233/asy-221801","url":null,"abstract":"This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α → 0. The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45158176","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}
In this paper we consider a model of laminated beams combining viscoelastic damping and strong time-delayed damping. The global well-posedness is proved by using the theory of semigroups of linear operators. We prove the lack of exponential stability when the speed wave propagations are not equal. In fact, we show in this situation, that the system goes to zero polynomially with rate t − 1 / 2 . On the other hand, by constructing some suitable multipliers, we establish that the energy decays exponentially provided the equal-speed wave propagations hold.
{"title":"Stability analysis of laminated beams with Kelvin–Voigt damping and strong time delay","authors":"C. Nonato, C. Raposo, B. Feng, A. Ramos","doi":"10.3233/asy-221802","DOIUrl":"https://doi.org/10.3233/asy-221802","url":null,"abstract":"In this paper we consider a model of laminated beams combining viscoelastic damping and strong time-delayed damping. The global well-posedness is proved by using the theory of semigroups of linear operators. We prove the lack of exponential stability when the speed wave propagations are not equal. In fact, we show in this situation, that the system goes to zero polynomially with rate t − 1 / 2 . On the other hand, by constructing some suitable multipliers, we establish that the energy decays exponentially provided the equal-speed wave propagations hold.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47585851","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}
Fakhrielddine Bader, M. Bendahmane, Mazen Saad, Raafat Talhouk
We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.
{"title":"Microscopic tridomain model of electrical activity in the heart with dynamical gap junctions. Part 2 – Derivation of the macroscopic tridomain model by unfolding homogenization method","authors":"Fakhrielddine Bader, M. Bendahmane, Mazen Saad, Raafat Talhouk","doi":"10.3233/ASY-221804","DOIUrl":"https://doi.org/10.3233/ASY-221804","url":null,"abstract":"We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math. 179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness’s result.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49276814","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}
This paper is concerned with the long time behavior of solutions for a non-autonomous reaction-diffusion equations with anomalous diffusion. Under suitable assumptions on nonlinearity and external force, the global well-posedness has been studied. Then the pullback attractors in L 2 ( Ω ) and H 0 α ( Ω ) ( 0 < α < 1) have been achieved with a restriction on the growth order of nonlinearity as 2 ⩽ p ⩽ 2 ( n − α ) n − 2 α . The results presented can be seen as the extension for classical theory of infinite dimensional dynamical system to the fractional diffusion equations.
{"title":"Dynamics and regularity for non-autonomous reaction-diffusion equations with anomalous diffusion","authors":"Xingjie Yan, Shubin Wang, Xinguang Yang, Junzhao Zhang","doi":"10.3233/asy-221800","DOIUrl":"https://doi.org/10.3233/asy-221800","url":null,"abstract":"This paper is concerned with the long time behavior of solutions for a non-autonomous reaction-diffusion equations with anomalous diffusion. Under suitable assumptions on nonlinearity and external force, the global well-posedness has been studied. Then the pullback attractors in L 2 ( Ω ) and H 0 α ( Ω ) ( 0 < α < 1) have been achieved with a restriction on the growth order of nonlinearity as 2 ⩽ p ⩽ 2 ( n − α ) n − 2 α . The results presented can be seen as the extension for classical theory of infinite dimensional dynamical system to the fractional diffusion equations.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48241273","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}
We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p-Laplace equation − Δ p u + ( − Δ ) p s u = c ( x ) | u | p − 2 u in Ω , where Ω ⊂ R N is an open set, p ∈ ( 1 , ∞ ), s ∈ ( 0 , 1 ) and c ∈ C ( Ω ‾ ).
我们建立了混合局部和非局部p-拉普拉斯方程−Δ p u +(−Δ) p s u = c (x) | u | p−2u在Ω中的弱解的一个强极大值原理，其中Ω∧R N是一个开集，p∈(1，∞)，s∈(0,1)，c∈c (Ω)。
{"title":"A strong maximum principle for mixed local and nonlocal p-Laplace equations","authors":"Bin Shang, Chao Zhang","doi":"10.3233/asy-221803","DOIUrl":"https://doi.org/10.3233/asy-221803","url":null,"abstract":"We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p-Laplace equation − Δ p u + ( − Δ ) p s u = c ( x ) | u | p − 2 u in Ω , where Ω ⊂ R N is an open set, p ∈ ( 1 , ∞ ), s ∈ ( 0 , 1 ) and c ∈ C ( Ω ‾ ).","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42354316","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}
A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics.
{"title":"Collective motion driven by nutrient consumption","authors":"P. Jabin, B. Perthame","doi":"10.3233/asy-221820","DOIUrl":"https://doi.org/10.3233/asy-221820","url":null,"abstract":"A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42609196","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}
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