We obtain new results about the high-energy distribution of resonances for the one-dimensional Schrödinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients, refining methods used in (J. Differential Equations 137(2) (1997) 251–272) and (J. Funct. Anal. 178(2) (2000) 396–420).
{"title":"Singularities and asymptotic distribution of resonances for Schrödinger operators in one dimension","authors":"T. J. Christiansen, T. Cunningham","doi":"10.3233/asy-241928","DOIUrl":"https://doi.org/10.3233/asy-241928","url":null,"abstract":"We obtain new results about the high-energy distribution of resonances for the one-dimensional Schrödinger operator. Our primary result is an upper bound on the density of resonances above any logarithmic curve in terms of the singular support of the potential. We also prove results about the distribution of resonances in sectors away from the real axis, and construct a class of potentials producing multiple sequences of resonances along distinct logarithmic curves, explicitly calculating the asymptotic location of these resonances. The results are unified by the use of an integral representation of the reflection coefficients, refining methods used in (J. Differential Equations 137(2) (1997) 251–272) and (J. Funct. Anal. 178(2) (2000) 396–420).","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141926123","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 the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ6 model in dimension 1+1. We construct a sequence of approximate solutions (ϕk(v,t,x))k∈N⩾2 for this model to understand the effects of thecollision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ6 model.
在本文中，我们考虑了维数为 1+1 的非线性波方程（称为 ϕ6 模型）中两个低速 v 扭结碰撞的弹性和稳定性问题。我们构建了该模型的近似解 (ϕk(v,t,x))k∈N⩾2 序列，以了解碰撞对大时间间隔内每个孤子运动的影响。该构造使用了一种新的渐近方法，它不仅限于ϕ6 模型。
{"title":"Approximate kink-kink solutions for the ϕ6 model in the low-speed limit","authors":"Abdon Moutinho","doi":"10.3233/asy-241917","DOIUrl":"https://doi.org/10.3233/asy-241917","url":null,"abstract":"In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ6 model in dimension 1+1. We construct a sequence of approximate solutions (ϕk(v,t,x))k∈N⩾2 for this model to understand the effects of thecollision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ6 model.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198510","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 article, our objective is to explore a Cahn–Hilliard system with a proliferation term, particularly relevant in biological contexts, with Neumann boundary conditions. We commence our investigation by establishing the boundedness of the average values of the local cell density u and the temperature H. This observation suggests that the solution (u,H) either persists globally in time or experiences finite-time blow-up. Subsequently, we prove the convergence of u to 1 and H to 0 as time approaches infinity. Finally, we bolster our theoretical findings with numerical simulations.
在本文中，我们的目标是探索一个带有增殖项的卡恩-希利亚德系统，该系统与生物背景特别相关，并具有诺伊曼边界条件。我们首先确定了局部细胞密度 u 和温度 H 平均值的有界性。随后，我们证明了当时间接近无穷大时，u 趋近于 1，H 趋近于 0。最后，我们用数值模拟来支持我们的理论发现。
{"title":"Cahn–Hilliard system with proliferation term","authors":"Aymard Christbert Nimi, Franck Davhys Reval Langa","doi":"10.3233/asy-241915","DOIUrl":"https://doi.org/10.3233/asy-241915","url":null,"abstract":"In this article, our objective is to explore a Cahn–Hilliard system with a proliferation term, particularly relevant in biological contexts, with Neumann boundary conditions. We commence our investigation by establishing the boundedness of the average values of the local cell density u and the temperature H. This observation suggests that the solution (u,H) either persists globally in time or experiences finite-time blow-up. Subsequently, we prove the convergence of u to 1 and H to 0 as time approaches infinity. Finally, we bolster our theoretical findings with numerical simulations.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198667","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 following Hamilton–Choquard type elliptic system: −Δu+u=(Iα∗F(v))f(v),x∈R2,−Δv+v=(Iβ∗F(u))f(u),x∈R2, where Iα and Iβ are Riesz potentials, f:R→R possessing critical exponential growth at infinity and F(t)=∫0tf(s)ds. Without the classic Ambrosetti–Rabinowitz conditionand strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.
本文研究以下汉密尔顿-觊觎型椭圆系统：-Δu+u=(Iα∗F(v))f(v),x∈R2,-Δv+v=(Iβ∗F(u))f(u),x∈R2，其中 Iα 和 Iβ 是里兹势，f:R→R 在无穷远处具有临界指数增长，F(t)=∫0tf(s)ds。在不考虑经典的 Ambrosetti-Rabinowitz 条件和 f 的严格单调性条件的情况下，我们将研究上述系统的基态解的存在性。该系统的强不确定性特征，加上卷积项和临界指数增长，使得该问题的研究既有趣又具有挑战性。在适当辅助系统的帮助下，我们采用近似方案和非内哈里流形方法，通过精细阈值控制最小值水平，成功地恢复了临界问题的紧凑性。通过集中紧凑性论证和一些详细估计，最终确定了基态解的存在性。
{"title":"Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth","authors":"Minlan Guan, Lizhen Lai, Boxue Liu, Dongdong Qin","doi":"10.3233/asy-241916","DOIUrl":"https://doi.org/10.3233/asy-241916","url":null,"abstract":"In this paper, we study the following Hamilton–Choquard type elliptic system: −Δu+u=(Iα∗F(v))f(v),x∈R2,−Δv+v=(Iβ∗F(u))f(u),x∈R2, where Iα and Iβ are Riesz potentials, f:R→R possessing critical exponential growth at infinity and F(t)=∫0tf(s)ds. Without the classic Ambrosetti–Rabinowitz conditionand strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198666","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 present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in LN(Ω), with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H01(Ω). However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.
{"title":"An elliptic problem in dimension N with a varying drift term bounded in LN","authors":"Juan Casado-Díaz","doi":"10.3233/asy-241914","DOIUrl":"https://doi.org/10.3233/asy-241914","url":null,"abstract":"The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in LN(Ω), with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H01(Ω). However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141932520","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 presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wienerfunctional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H<1/2, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.
{"title":"New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion","authors":"Akihiko Takahashi, Toshihiro Yamada","doi":"10.3233/asy-241910","DOIUrl":"https://doi.org/10.3233/asy-241910","url":null,"abstract":"This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wienerfunctional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H<1/2, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198668","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}
Yongqing Zhao, Wenjun Liu, Guangying Lv, Yuepeng Wang
In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H2-strong solution of the assimilation system converges exponentially to the reference solution in the sense of L2 as t→∞. We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.
{"title":"Continuous data assimilation for the three dimensional primitive equations with magnetic field","authors":"Yongqing Zhao, Wenjun Liu, Guangying Lv, Yuepeng Wang","doi":"10.3233/asy-241912","DOIUrl":"https://doi.org/10.3233/asy-241912","url":null,"abstract":"In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H2-strong solution of the assimilation system converges exponentially to the reference solution in the sense of L2 as t→∞. We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198669","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 work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω, for some C > 0, ∀ t , s ∈ R, p < σ i ⩽ p ∗ : = N p / ( N − p ), i = 1 , 2, and 1 < p < N. We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.
在这项工作中，我们考虑一个函数 I ：W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R 的形式 I ( u , v ) = 1 p ∫ Ω ( |∇ u | p + |∇ v | p ) d x - ∫ Ω H ( x 、u ( x ) , v ( x ) ) d x 其中 Ω ⊂ R N 是一个光滑有界域，max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | }⩽ C ( 1 + | s | σ 1 - 1 + | t | σ 2 - 1 ) a.e. x∈ Ω, 对于某个 C > 0, ∀ t , s∈ R, p < σ i ⩽ p∗ : = N p / ( N - p ), i = 1 , 2, 且 1 < p < N。我们证明 C 0 1 ( Ω ) × C 0 1 ( Ω ) 拓扑中的局部最小值就是 W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) 拓扑中的局部最小值。这一结果的一个重要应用与一类凹凸型非线性系统的解的多重性问题有关。
{"title":"W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers","authors":"João Pablo P. Da Silva","doi":"10.3233/asy-241911","DOIUrl":"https://doi.org/10.3233/asy-241911","url":null,"abstract":"In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω, for some C > 0, ∀ t , s ∈ R, p < σ i ⩽ p ∗ : = N p / ( N − p ), i = 1 , 2, and 1 < p < N. We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140995918","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 the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.
在本文中，我们考虑广义的准薛定谔方程 - div ( g 2 ( u )∇ u )+ g ( u ) g ′ ( u ) |∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N ，其中 V 和 h 在 x i 中是周期性的，1 ⩽ i ⩽ N。通过使用变分法，我们证明了基态解的存在，即具有最小可能能量的非微分解。
{"title":"Ground state solutions for generalized quasilinear Schrödinger equations","authors":"Xiang-Dong Fang, Zhi-Qing Han","doi":"10.3233/asy-241913","DOIUrl":"https://doi.org/10.3233/asy-241913","url":null,"abstract":"In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141018125","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 inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n⩾3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.
我们研究了在维数为 n⩾3 且边界相连的保角横向各向异性黎曼流形上具有荷尔德连续磁势和连续电势的磁薛定谔算子的逆边界问题。只要横向流形上的大地 X 射线变换是注入式的，就能根据流形边界上的部分考奇数据建立磁场和电势的全局唯一性结果。
{"title":"Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds","authors":"Salem Selim, Lili Yan","doi":"10.3233/asy-241909","DOIUrl":"https://doi.org/10.3233/asy-241909","url":null,"abstract":"We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n⩾3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508377","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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