We consider a double phase (unbalanced growth) Dirichlet problem with a Carathéodory reaction f ( z , x ) which is superlinear in x but without satisfying the AR-condition. Using the symmetric mountain pass theorem, we produce a whole sequence of distinct bounded solutions which diverge to infinity.
{"title":"Divergent sequence of nontrivial solutions for superlinear double phase problems","authors":"Nikolaos S. Papageorgiou, C. Vetro, F. Vetro","doi":"10.3233/asy-231830","DOIUrl":"https://doi.org/10.3233/asy-231830","url":null,"abstract":"We consider a double phase (unbalanced growth) Dirichlet problem with a Carathéodory reaction f ( z , x ) which is superlinear in x but without satisfying the AR-condition. Using the symmetric mountain pass theorem, we produce a whole sequence of distinct bounded solutions which diverge to infinity.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43753457","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 the motion of spherical particles in the whole space R 3 filled with a viscous fluid. We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.
{"title":"Global solutions to coupled (Navier-)Stokes Newton systems in R 3","authors":"M. Hillairet, L. Sabbagh","doi":"10.3233/asy-221790","DOIUrl":"https://doi.org/10.3233/asy-221790","url":null,"abstract":"We consider the motion of spherical particles in the whole space R 3 filled with a viscous fluid. We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135383663","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}
Jiang, Yan, Liu, Hongyu, Zhang, Jiachuan, Zhang, Kai
Consider the transmission eigenvalue problem for u ∈ H 1 ( Ω ) and v ∈ H 1 ( Ω ): ∇ · ( σ ∇ u ) + k 2 n 2 u = 0 in Ω , Δ v + k 2 v = 0 in Ω , u = v , σ ∂ u ∂ ν = ∂ v ∂ ν on ∂ Ω , where Ω is a ball in R N , N = 2 , 3. If σ and n are both radially symmetric, namely they are functions of the radial parameter r only, we show that there exists a sequence of transmission eigenfunctions { u m , v m } m ∈ N associated with k m → + ∞ as m → + ∞ such that the L 2 -energies of v m ’s are concentrated around ∂ Ω. If σ and n are both constant, we show the existence of transmission eigenfunctions { u j , v j } j ∈ N such that both u j and v j are localized around ∂ Ω. Our results extend the recent studies in (SIAM J. Imaging Sci. 14 (2021), 946–975; Chow et al.). Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.
认为《传输eigenvalue问题for u H∈v H∈(Ω)和1(Ω):∇·(σ∇u) k + 2 n u = 0在Ω,Δv + k 2 v = 0在Ω,u = v,σ∂u∂ν=∂v∂νon∂Ω,Ω哪儿是一个球在R n, n = 2, 3。如果σ和n都radially symmetric, namely,他们是径向functions of the参数r才,我们的节目有exists a序列的传输eigenfunctions {u, v m, m∈n (associated with k m→+∞(美国)→+∞L 2 -energies》如此那v m ' s是∂周围深Ω。如果σ和n都康斯坦,我们存在》节目传输eigenfunctions {u v j, j} j∈n如此j j这两者u和v是∂周围localizedΩ。我们最近的扩展研究(暹罗J. Imaging Sci. 14(2021)， 946—975;周和艾尔。无论是numerics影响》,我们也discuss parameters媒介,namelyσ和n,几何上的传输eigenfunctions之模式。
{"title":"Boundary localization of transmission eigenfunctions in spherically stratified media","authors":"Jiang, Yan, Liu, Hongyu, Zhang, Jiachuan, Zhang, Kai","doi":"10.3233/asy-221794","DOIUrl":"https://doi.org/10.3233/asy-221794","url":null,"abstract":"Consider the transmission eigenvalue problem for u ∈ H 1 ( Ω ) and v ∈ H 1 ( Ω ): ∇ · ( σ ∇ u ) + k 2 n 2 u = 0 in Ω , Δ v + k 2 v = 0 in Ω , u = v , σ ∂ u ∂ ν = ∂ v ∂ ν on ∂ Ω , where Ω is a ball in R N , N = 2 , 3. If σ and n are both radially symmetric, namely they are functions of the radial parameter r only, we show that there exists a sequence of transmission eigenfunctions { u m , v m } m ∈ N associated with k m → + ∞ as m → + ∞ such that the L 2 -energies of v m ’s are concentrated around ∂ Ω. If σ and n are both constant, we show the existence of transmission eigenfunctions { u j , v j } j ∈ N such that both u j and v j are localized around ∂ Ω. Our results extend the recent studies in (SIAM J. Imaging Sci. 14 (2021), 946–975; Chow et al.). Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135383283","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 explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.
{"title":"Singularity extraction for elliptic equations with coefficients with jumps and Dirac sources","authors":"Eya Bejaoui, F. Ben Belgacem","doi":"10.3233/asy-221824","DOIUrl":"https://doi.org/10.3233/asy-221824","url":null,"abstract":"We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42309073","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}
There is a narrow but hidden link between optimal control theory and the so-called Tikhonov regularization method. In fact, the small coefficient representing the marginal cost of the control can be interpreted as the regularization parameter in a Tikhonov method as far as there exists an exact control. This strategy enables one to adjust the cost function in the optimal control model in order to define the exact control which minimizes a given functional involving both the control but also the state variables during the control process. The goal of this paper is to suggest a method which gives a simple way to characterize and compute the exact control corresponding to the minimum of a given cost functional as said above. It appears as an extension of the phase control which is a finite dimensional version of the HUM control of J.L. Lions but for partial differential equations.
{"title":"Asymptotic method and transient terms in exact controls","authors":"P. Destuynder","doi":"10.3233/asy-231829","DOIUrl":"https://doi.org/10.3233/asy-231829","url":null,"abstract":"There is a narrow but hidden link between optimal control theory and the so-called Tikhonov regularization method. In fact, the small coefficient representing the marginal cost of the control can be interpreted as the regularization parameter in a Tikhonov method as far as there exists an exact control. This strategy enables one to adjust the cost function in the optimal control model in order to define the exact control which minimizes a given functional involving both the control but also the state variables during the control process. The goal of this paper is to suggest a method which gives a simple way to characterize and compute the exact control corresponding to the minimum of a given cost functional as said above. It appears as an extension of the phase control which is a finite dimensional version of the HUM control of J.L. Lions but for partial differential equations.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44561125","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 a class of the ( p , q ) Kirchhoff type problems with convolution term in R N . With the appropriate assumptions on potential function V and convolution term f, together with the penalization techniques, Morse iterative method and variational method, the existence and multiplicity of multi-bump solutions are obtained for this problem. In some sense, our results also generalize some known results.
{"title":"Existence and multiplicity of multi-bump solutions for the double phase Kirchhoff problems with convolution term in R N","authors":"Shuaishuai Liang, S. Shi","doi":"10.3233/asy-231827","DOIUrl":"https://doi.org/10.3233/asy-231827","url":null,"abstract":"In this paper, we study a class of the ( p , q ) Kirchhoff type problems with convolution term in R N . With the appropriate assumptions on potential function V and convolution term f, together with the penalization techniques, Morse iterative method and variational method, the existence and multiplicity of multi-bump solutions are obtained for this problem. In some sense, our results also generalize some known results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42937661","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. Ramos, A. Araujo, A. Campelo, M. Freitas, L. S. Veras
In this paper, we are interested in studying the well-posedness, optimal polynomial stability, and the lack of exponential stability for a class of thermoelastic system of Reissner–Mindlin–Timoshenko plates with structural damping, that is, with the dissipation of Kelvin–Voigt type on the equations for the rotation angles. We also consider the thermal effect with thermal variables described by Fourier’s law of heat conduction.
{"title":"Polynomial stabilization for thermoelastic Reissner–Mindlin–Timoshenko plates with structural damping","authors":"A. Ramos, A. Araujo, A. Campelo, M. Freitas, L. S. Veras","doi":"10.3233/asy-231826","DOIUrl":"https://doi.org/10.3233/asy-231826","url":null,"abstract":"In this paper, we are interested in studying the well-posedness, optimal polynomial stability, and the lack of exponential stability for a class of thermoelastic system of Reissner–Mindlin–Timoshenko plates with structural damping, that is, with the dissipation of Kelvin–Voigt type on the equations for the rotation angles. We also consider the thermal effect with thermal variables described by Fourier’s law of heat conduction.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41340939","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}
Hussein Raad, L. Cherfils, C. Allery, R. Guillevin
The aim of this paper is to first study a Cahn-Hilliard model for brain lactate kinetics with a control function. This control allows for optimal treatment administered to ill patients suffering from glioma, in order to reduce their brain lactate concentrations, and thereby to slow down the tumor growth. We establish the well-posedness of the problem and the continuity of the control-to-state mapping, the existence of a minimizer of the objective functional, and its Fréchet differentiability in suitable Banach spaces with respect to the control and with respect to time. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. In the second part of the paper, we illustrate our theoretical results with numerical simulations using MRI data from the University Hospital of Poitiers.
{"title":"Optimal control of a model for brain lactate kinetics","authors":"Hussein Raad, L. Cherfils, C. Allery, R. Guillevin","doi":"10.3233/asy-221823","DOIUrl":"https://doi.org/10.3233/asy-221823","url":null,"abstract":"The aim of this paper is to first study a Cahn-Hilliard model for brain lactate kinetics with a control function. This control allows for optimal treatment administered to ill patients suffering from glioma, in order to reduce their brain lactate concentrations, and thereby to slow down the tumor growth. We establish the well-posedness of the problem and the continuity of the control-to-state mapping, the existence of a minimizer of the objective functional, and its Fréchet differentiability in suitable Banach spaces with respect to the control and with respect to time. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. In the second part of the paper, we illustrate our theoretical results with numerical simulations using MRI data from the University Hospital of Poitiers.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41880950","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 will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution.
{"title":"Cahn–Hilliard equation with regularization term","authors":"Rim Mheich","doi":"10.3233/asy-221821","DOIUrl":"https://doi.org/10.3233/asy-221821","url":null,"abstract":"We will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46091030","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 existence results of a quasilinear elliptic problem involving the 1-biharmonic operator in R N , whose nonlinearity satisfies appropriate conditions. The existence theorem is proved through a new version of the Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais–Smale one.
{"title":"Existence of nontrivial solution for quasilinear equations involving the 1-biharmonic operator","authors":"Huo Tao, Lin Li, Xiao-Qiong Yang","doi":"10.3233/asy-221822","DOIUrl":"https://doi.org/10.3233/asy-221822","url":null,"abstract":"In this paper, we study the existence results of a quasilinear elliptic problem involving the 1-biharmonic operator in R N , whose nonlinearity satisfies appropriate conditions. The existence theorem is proved through a new version of the Mountain Pass Theorem to locally Lipschitz functionals, where it is considered the Cerami compactness condition rather than the Palais–Smale one.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45847313","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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