In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain Ω − juxtaposed with a planar thin layer Ω + δ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer Ω + δ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn’t take into account any more the thin layer Ω + δ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem.
{"title":"Approximation of Dirichlet-to-Neumann operator for a planar thin layer and stabilization in the framework of couple stress elasticity with voids","authors":"Athmane Abdallaoui, A. Kelleche","doi":"10.3233/asy-231886","DOIUrl":"https://doi.org/10.3233/asy-231886","url":null,"abstract":"In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain Ω − juxtaposed with a planar thin layer Ω + δ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer Ω + δ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn’t take into account any more the thin layer Ω + δ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138588081","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 study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.
{"title":"Semiclassical WKB problem for the non-self-adjoint Dirac operator with a multi-humped decaying potential","authors":"Nicholas Hatzizisis, Spyridon Kamvissis","doi":"10.3233/asy-231885","DOIUrl":"https://doi.org/10.3233/asy-231885","url":null,"abstract":"In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139665270","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 analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H1-convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.
{"title":"Asymptotic decay towards steady states of solutions to very fast and singular diffusion equations","authors":"Georgy Kitavtsev, Roman M. Taranets","doi":"10.3233/asy-231884","DOIUrl":"https://doi.org/10.3233/asy-231884","url":null,"abstract":"We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H1-convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139664160","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}
Victor Cabanillas Zannini, Teófanes Quispe Méndez, A.J.A. Ramos
This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.
{"title":"Optimal stability for laminated beams with Kelvin–Voigt damping and Fourier’s law","authors":"Victor Cabanillas Zannini, Teófanes Quispe Méndez, A.J.A. Ramos","doi":"10.3233/asy-231883","DOIUrl":"https://doi.org/10.3233/asy-231883","url":null,"abstract":"This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083437","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 dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of theglobal attractor.
{"title":"Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term","authors":"Yuming Qin, Hongli Wang, Bin Yang","doi":"10.3233/asy-231881","DOIUrl":"https://doi.org/10.3233/asy-231881","url":null,"abstract":"This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of theglobal attractor.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083381","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 prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences (Em+)m=0∞, (Em−)m=0∞, satisfying a two-term asymptotic formula with error estimate of the form O(m−1/4), when m tends to infinity.
我们证明，当 m 趋于无穷大时，非对称量子拉比模型的频谱由两个特征值序列 (Em+)m=0∞, (Em-)m=0∞ 组成，满足两期渐近公式，误差估计为 O(m-1/4)。
{"title":"Behaviour of large eigenvalues for the asymmetric quantum Rabi model","authors":"Mirna Charif, Ahmad Fino, Lech Zielinski","doi":"10.3233/asy-231875","DOIUrl":"https://doi.org/10.3233/asy-231875","url":null,"abstract":"We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences (Em+)m=0∞, (Em−)m=0∞, satisfying a two-term asymptotic formula with error estimate of the form O(m−1/4), when m tends to infinity.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138561049","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 analyze a non-linear elliptic boundary value problem that involves (p,q) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in (0,∞) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving (p,q) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.
{"title":"On a class of infinite semipositone problems for (p,q) Laplace operator","authors":"R. Dhanya, Sarbani Pramanik, R. Harish","doi":"10.3233/asy-231880","DOIUrl":"https://doi.org/10.3233/asy-231880","url":null,"abstract":"We analyze a non-linear elliptic boundary value problem that involves (p,q) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in (0,∞) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving (p,q) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101875","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 a bounded cylinder with a rough interface we study the asymptotic behaviour of the spectrum and its associated eigenspaces for a stationary heat propagation problem. The main novelty concerns the proof of the uniform a priori estimates for the eigenvalues. In fact, due to the peculiar geometry of the domain, standard techniques do not apply and a suitable new approach is developed.
{"title":"Homogenization of an eigenvalue problem through rough surfaces","authors":"J. Avila, Sara Monsurrò, F. Raimondi","doi":"10.3233/asy-231882","DOIUrl":"https://doi.org/10.3233/asy-231882","url":null,"abstract":"In a bounded cylinder with a rough interface we study the asymptotic behaviour of the spectrum and its associated eigenspaces for a stationary heat propagation problem. The main novelty concerns the proof of the uniform a priori estimates for the eigenvalues. In fact, due to the peculiar geometry of the domain, standard techniques do not apply and a suitable new approach is developed.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139273624","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 are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter κ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ regardless of the size of initial data.
{"title":"Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects","authors":"V. Angulo-Castillo, L.C.F. Ferreira, L. Kosloff","doi":"10.3233/asy-231879","DOIUrl":"https://doi.org/10.3233/asy-231879","url":null,"abstract":"We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter κ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ regardless of the size of initial data.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136351520","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 first show that under a suitable balanced repartition of the mixed controls within the system, Kalman’s rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls.
{"title":"Approximate mixed synchronization by groups for a coupled system of wave equations","authors":"Tatsien Li, Bopeng Rao","doi":"10.3233/asy-231865","DOIUrl":"https://doi.org/10.3233/asy-231865","url":null,"abstract":"We first show that under a suitable balanced repartition of the mixed controls within the system, Kalman’s rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092429","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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