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":"3 1","pages":""},"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 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":"69 1","pages":""},"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}
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":"32 1","pages":""},"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}
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":"1 1","pages":""},"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":"75 1","pages":""},"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":"2013 1","pages":""},"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}
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":"72 1","pages":""},"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}
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":"31 1","pages":""},"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}
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":"72 1","pages":""},"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}
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":"57 1","pages":""},"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}
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