The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p=2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.
{"title":"Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction","authors":"Rama Rawat, Haripada Roy, Prosenjit Roy","doi":"10.3233/asy-241907","DOIUrl":"https://doi.org/10.3233/asy-241907","url":null,"abstract":"The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p=2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141519164","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}
The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions.
{"title":"Asymptotic behaviour of some anisotropic problems","authors":"Michel Chipot","doi":"10.3233/asy-241906","DOIUrl":"https://doi.org/10.3233/asy-241906","url":null,"abstract":"The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140669113","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 present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.
{"title":"An optimal control problem for diffusion–precipitation model1","authors":"A. Kundu, H. S. Mahato","doi":"10.3233/asy-241905","DOIUrl":"https://doi.org/10.3233/asy-241905","url":null,"abstract":"We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140670868","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}
Ting Deng, Marco Squassina, Jianjun Zhang, Xuexiu Zhong
We are concerned with solutions of the following quasilinear Schrödinger equations −div(φ2(u)∇u)+φ(u)φ′(u)|∇u|2+λu=f(u),x∈RN with prescribed mass ∫RNu2dx=c, where N⩾3, c>0, λ∈R appears as the Lagrange multiplier and φ∈C1(R,R+). The nonlinearity f∈C(R,R) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.
我们关注的是以下准线性薛定谔方程的解-div(φ2(u)∇u)+φ(u)φ′(u)|∇u|2+λu=f(u),x∈RN，其中 N⩾3，c>；0，λ∈R 作为拉格朗日乘数出现，φ∈C1(R,R+)。允许非线性 f∈C(R,R) 在原点和无穷远处为质量次临界、质量临界和质量超临界。通过对偶方法、定点索引和全局分支方法，我们确定了上述问题的归一化解的存在性。这些结果将 L. Jeanjean、J. J. Zhang 和 X.X. Zhong 以前的结果扩展到了准线性情况。
{"title":"Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity","authors":"Ting Deng, Marco Squassina, Jianjun Zhang, Xuexiu Zhong","doi":"10.3233/asy-241908","DOIUrl":"https://doi.org/10.3233/asy-241908","url":null,"abstract":"We are concerned with solutions of the following quasilinear Schrödinger equations −div(φ2(u)∇u)+φ(u)φ′(u)|∇u|2+λu=f(u),x∈RN with prescribed mass ∫RNu2dx=c, where N⩾3, c>0, λ∈R appears as the Lagrange multiplier and φ∈C1(R,R+). The nonlinearity f∈C(R,R) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508378","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 the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ = 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β. Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.
本文研究了具有分数记忆和分数扩散的布森斯克系统的局部和全局自相似解的存在性 u t + u - ∇ u + ∇ p + ν ( - Δ ) β u = θ f , x ∈ R n 、t > 0 , θ t + u -∇ θ + g α ∗ ( - Δ ) γ θ = 0 , x∈ R n , t > 0 , div u = 0 , x∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x∈ R n , 其中 g α ( t ) = t α - 1 Γ ( α ) 。存在性结果是在伪度量空间框架下得到的。我们发现，全局解的存在性和自相似性深受记忆和分数扩散之间关系的影响。事实上，只有当 γ = ( α + 1 ) β 时，我们才能得到全局解的存在性和自相似性。此外，我们还证明了全局解的稳定性结果和渐近自相似解的存在性。
{"title":"On the Boussinesq system with fractional memory in pseudo-measure spaces","authors":"Felipe Poblete, Clessius Silva, A. Viana","doi":"10.3233/asy-241904","DOIUrl":"https://doi.org/10.3233/asy-241904","url":null,"abstract":"This paper studies the existence of local and global self-similar solutions for a Boussinesq system with fractional memory and fractional diffusions u t + u · ∇ u + ∇ p + ν ( − Δ ) β u = θ f , x ∈ R n , t > 0 , θ t + u · ∇ θ + g α ∗ ( − Δ ) γ θ = 0 , x ∈ R n , t > 0 , div u = 0 , x ∈ R n , t > 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 , x ∈ R n , where g α ( t ) = t α − 1 Γ ( α ) . The existence results are obtained in the framework of pseudo-measure spaces. We find that the existence and self-similarity of global solutions is strongly influenced by the relationship among the memory and the fractional diffusions. Indeed, we obtain the existence and self-similarity of global solutions only when γ = ( α + 1 ) β. Moreover, we prove a stability result for the global solutions and the existence of asymptotically self-similar solutions.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140381384","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 fractional Schrödinger–Poisson system with p-Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.
本文研究了一个具有 p 拉普拉斯的分数薛定谔-泊松系统。通过使用一些缩放变换和截断技术，得到了山口级 Palais-Smale 序列的有界性。因此，得到了系统的非三维解的存在性。
{"title":"Non-trivial solutions for the fractional Schrödinger–Poisson system with p-Laplacian","authors":"Chungen Liu, Yuyou Zhong, Jiabin Zuo","doi":"10.3233/asy-241903","DOIUrl":"https://doi.org/10.3233/asy-241903","url":null,"abstract":"In this paper, we study a fractional Schrödinger–Poisson system with p-Laplacian. By using some scaling transformation and cut-off technique, the boundedness of the Palais–Smale sequences at the mountain pass level is gotten. As a result, the existence of non-trivial solutions for the system is obtained.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140381714","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 derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓSG and ℓchiral, and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓSG, ℓchiral, ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator.
{"title":"Effective medium theory for second-gradient elasticity with chirality","authors":"Grigor Nika, Adrian Muntean","doi":"10.3233/asy-241902","DOIUrl":"https://doi.org/10.3233/asy-241902","url":null,"abstract":"We derive effective models for a heterogeneous second-gradient elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓSG and ℓchiral, and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓSG, ℓchiral, ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors’ structure rely on a suitable use of the periodic unfolding operator.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140802820","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 study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021)), which means the optimal convergence rate is also O(ε).
{"title":"Optimal convergence rate for homogenization of convex Hamilton–Jacobi equations in the periodic spatial-temporal environment","authors":"Hoang Nguyen-Tien","doi":"10.3233/asy-241898","DOIUrl":"https://doi.org/10.3233/asy-241898","url":null,"abstract":"We study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021)), which means the optimal convergence rate is also O(ε).","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140802628","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}
Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r∗ of the initial data in H1(R3). Motivated by this work, we focus on characterizing thelarge-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting methods. In particular, for the case −n2<r∗⩽1, we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case.
{"title":"Decay characterization of solutions to incompressible Navier–Stokes–Voigt equations","authors":"Jitao Liu, Shasha Wang, Wen-Qing Xu","doi":"10.3233/asy-241900","DOIUrl":"https://doi.org/10.3233/asy-241900","url":null,"abstract":"Recently, Niche [J. Differential Equations, 260 (2016), 4440–4453] established upper bounds on the decay rates of solutions to the 3D incompressible Navier–Stokes–Voigt equations in terms of the decay character r∗ of the initial data in H1(R3). Motivated by this work, we focus on characterizing thelarge-time behavior of all space-time derivatives of the solutions for the 2D case and establish upper bounds and lower bounds on their decay rates by making use of the decay character and Fourier splitting methods. In particular, for the case −n2<r∗⩽1, we verify the optimality of the upper bounds, which is new to the best of our knowledge. Similar improved decay results are also true for the 3D case.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314983","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 Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.
{"title":"Global well-posedness and L 2 decay estimate of smooth solutions for the 3-D incompressible Navier–Stokes–Allen–Cahn system","authors":"Wenwen Huo, Kaimin Teng, Chao Zhang","doi":"10.3233/asy-241901","DOIUrl":"https://doi.org/10.3233/asy-241901","url":null,"abstract":"We consider the Cauchy problem for the 3-D incompressible Navier–Stokes–Allen–Cahn system, which can effectively describe large deformations or topological deformations. Under the assumptions of small initial data, we study the global well-posedness and time-decay of solutions to such system by means of pure energy method and Fourier-splitting technique.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140077354","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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