Pub Date : 2024-11-29DOI: 10.1016/j.aml.2024.109406
Jinrui Zhang, Haijun Hu, Chuangxia Huang
In this paper, we consider a class of Fisher–KPP equations with delays appearing in both diffusion and reaction terms. By employing some differential inequality analyses, we prove that the delayed Fisher–KPP equation possesses a pair of quasi-upper and quasi-lower solutions which have absolutely continuous derivatives. Based on this, we apply the monotone iteration method and the Perron’s theorem to establish a sufficient criterion ensuring the existence of wave fronts. Our proof corrects the previous related research.
{"title":"Wave fronts for a class of delayed Fisher–KPP equations","authors":"Jinrui Zhang, Haijun Hu, Chuangxia Huang","doi":"10.1016/j.aml.2024.109406","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109406","url":null,"abstract":"In this paper, we consider a class of Fisher–KPP equations with delays appearing in both diffusion and reaction terms. By employing some differential inequality analyses, we prove that the delayed Fisher–KPP equation possesses a pair of quasi-upper and quasi-lower solutions which have absolutely continuous derivatives. Based on this, we apply the monotone iteration method and the Perron’s theorem to establish a sufficient criterion ensuring the existence of wave fronts. Our proof corrects the previous related research.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"3 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-28DOI: 10.1016/j.aml.2024.109404
Huanhuan Lu , Yufeng Zhang
By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the -independent part of the Lax pair. In the final section, the nonlinear Fourier transform pair will be used, after also taking into account the appropriate time evolution, for solving the Cauchy initial value problem of the three-spatial-dimensions KK equation with two temporal variables.
{"title":"Nonlocal ∂̄ formalism for the three-spatial-dimensions Kaup–Kuperschmidt equation with two temporal variables","authors":"Huanhuan Lu , Yufeng Zhang","doi":"10.1016/j.aml.2024.109404","DOIUrl":"10.1016/j.aml.2024.109404","url":null,"abstract":"<div><div>By complexifying the independent variables of the Kaup–Kuperschmidt (KK) equation, we derive the 4+2 integrable extension of the KK equation and its Lax pair. The construction of the associated nonlinear Fourier transform pair comprising both direct and inverse transforms is accomplished by conducting a spectral analysis of the <span><math><mi>t</mi></math></span>-independent part of the Lax pair. In the final section, the nonlinear Fourier transform pair will be used, after also taking into account the appropriate time evolution, for solving the Cauchy initial value problem of the three-spatial-dimensions KK equation with two temporal variables.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109404"},"PeriodicalIF":2.9,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-28DOI: 10.1016/j.aml.2024.109405
Reinhard Racke, Shuji Yoshikawa
We prove a decay estimate for solutions to a transmission problem for wave equations with different propagation speeds in an infinite waveguide. The problem represents the wave propagation in unbounded and layered composite materials in which different properties are given. It is a new composition of a waveguide problem and a transmission problem, motivated by a unit cell model for CFRP. The proof is based on splitting variables, partial eigenfunction expansions in the bounded cross section, and on an explicit Weyl type estimate for the eigenvalues.
{"title":"A transmission problem for wave equations in infinite waveguides","authors":"Reinhard Racke, Shuji Yoshikawa","doi":"10.1016/j.aml.2024.109405","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109405","url":null,"abstract":"We prove a decay estimate for solutions to a transmission problem for wave equations with different propagation speeds in an infinite waveguide. The problem represents the wave propagation in unbounded and layered composite materials in which different properties are given. It is a new composition of a waveguide problem and a transmission problem, motivated by a unit cell model for CFRP. The proof is based on splitting variables, partial eigenfunction expansions in the bounded cross section, and on an explicit Weyl type estimate for the eigenvalues.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"82 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1016/j.aml.2024.109382
Chaoyue Guan, Yuli Sun, Jing Niu
In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and 2D Allen–Cahn equations with Neumann and periodic boundary conditions. These experiments demonstrate the stability and accuracy in capturing phase transitions of the approach. Meanwhile, we also show the LSV methods can maintain physical properties such as energy dissipation and uniform boundedness. It is worth mentioning that we observe that the LSV methods achieve both optimal convergence and superconvergence as the numerical flux parameter is carefully selected.
{"title":"Legendre spectral volume methods for Allen–Cahn equations by the direct discontinuous Galerkin formula","authors":"Chaoyue Guan, Yuli Sun, Jing Niu","doi":"10.1016/j.aml.2024.109382","DOIUrl":"10.1016/j.aml.2024.109382","url":null,"abstract":"<div><div>In this paper, we introduce novel class of Legendre spectral volume (LSV) methods for solving Allen–Cahn equations. Each spectral volume (SV) is refined with <span><math><mi>k</mi></math></span> Gauss–Legendre points to define an arbitrary order control volume (CV). Moreover, the second derivative is handled using the direct discontinuous Galerkin (DDG) approach. Furthermore, four numerical experiments are detailed including 1D and 2D Allen–Cahn equations with Neumann and periodic boundary conditions. These experiments demonstrate the stability and accuracy in capturing phase transitions of the approach. Meanwhile, we also show the LSV methods can maintain physical properties such as energy dissipation and uniform boundedness. It is worth mentioning that we observe that the LSV methods achieve both optimal convergence and superconvergence as the numerical flux parameter is carefully selected.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109382"},"PeriodicalIF":2.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1016/j.aml.2024.109401
Huaijun Yang
In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in -norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the -norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error splitting technique used in the previous work. Finally, some numerical results are presented to confirm the theoretical analysis.
{"title":"A new error analysis of a linearized Euler Galerkin scheme for Schrödinger equation with cubic nonlinearity","authors":"Huaijun Yang","doi":"10.1016/j.aml.2024.109401","DOIUrl":"10.1016/j.aml.2024.109401","url":null,"abstract":"<div><div>In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error splitting technique used in the previous work. Finally, some numerical results are presented to confirm the theoretical analysis.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109401"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1016/j.aml.2024.109402
Wei Zhang , Jialing Zhang
<div><div>We consider this equation <span><span><span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mfenced><mrow><mi>p</mi><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mi>k</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. Here <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denotes the <span><math><mi>k</mi></math></span>th elementary symmetric function of the eigenvalues of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span>, and <span><span><span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><mo>∇</mo><mi>u</mi><mo>⊗</mo><mo>∇</mo><mi>u</mi><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>I</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mo>∇</mo><mi>u</mi></mrow></math></span> denotes the gradient of <span><math><mi>u</mi></math></span> and <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow></math></span> denotes the Hessian of <span><math><mi>u</mi></math></span>. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y
{"title":"Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations","authors":"Wei Zhang , Jialing Zhang","doi":"10.1016/j.aml.2024.109402","DOIUrl":"10.1016/j.aml.2024.109402","url":null,"abstract":"<div><div>We consider this equation <span><span><span><math><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mfenced><mrow><mi>p</mi><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mi>k</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>p</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. Here <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> denotes the <span><math><mi>k</mi></math></span>th elementary symmetric function of the eigenvalues of <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup></math></span>, and <span><span><span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><mo>∇</mo><mi>u</mi><mo>⊗</mo><mo>∇</mo><mi>u</mi><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mi>I</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mo>∇</mo><mi>u</mi></mrow></math></span> denotes the gradient of <span><math><mi>u</mi></math></span> and <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow></math></span> denotes the Hessian of <span><math><mi>u</mi></math></span>. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109402"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-24DOI: 10.1016/j.aml.2024.109391
Zhao Yang , Tao Chen , Sanyang Liu
In this paper, we show that collocation matrices of the Cauchy–Bernstein basis can be decomposed as products of a Cauchy–Vandermonde matrix and a block diagonal matrix. A useful application of this result is that the explicit expression of the determinant for the collocation matrices is presented. Consequently, an algorithm is provided to accurately compute the determinants. Numerical experiments confirm the high accuracy of the algorithm.
{"title":"On decomposition of collocation matrices for the Cauchy–Bernstein basis and applications","authors":"Zhao Yang , Tao Chen , Sanyang Liu","doi":"10.1016/j.aml.2024.109391","DOIUrl":"10.1016/j.aml.2024.109391","url":null,"abstract":"<div><div>In this paper, we show that collocation matrices of the Cauchy–Bernstein basis can be decomposed as products of a Cauchy–Vandermonde matrix and a block diagonal matrix. A useful application of this result is that the explicit expression of the determinant for the collocation matrices is presented. Consequently, an algorithm is provided to accurately compute the determinants. Numerical experiments confirm the high accuracy of the algorithm.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"163 ","pages":"Article 109391"},"PeriodicalIF":2.9,"publicationDate":"2024-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-23DOI: 10.1016/j.aml.2024.109393
Chunchi Liu , Yizhong Sun , Jiaping Yu
This paper investigates a fully parallel decoupled approach of the discrete stabilized finite element method for the time-dependent Stokes–Darcy problem. By introducing an auxiliary function, we rigorously demonstrate that the parallel algorithm is unconditionally stable.
{"title":"SAV unconditional stable estimate of parallel decoupled stabilized finite element algorithm for the fully mixed Stokes–Darcy problems","authors":"Chunchi Liu , Yizhong Sun , Jiaping Yu","doi":"10.1016/j.aml.2024.109393","DOIUrl":"10.1016/j.aml.2024.109393","url":null,"abstract":"<div><div>This paper investigates a fully parallel decoupled approach of the discrete stabilized finite element method for the time-dependent Stokes–Darcy problem. By introducing an auxiliary function, we rigorously demonstrate that the parallel algorithm is unconditionally stable.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109393"},"PeriodicalIF":2.9,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-23DOI: 10.1016/j.aml.2024.109390
Ziliang Yang , Jiabao Su , Mingzheng Sun
In this paper, we consider the following elliptic problem where the nonlinearity satisfies the Ambrosetti–Rabinowitz condition. Using an additional growth condition of at a bounded region, we can obtain five nontrivial solutions of by applying homological linking arguments and Morse theory.
本文考虑以下椭圆问题 -Δu=f(x,u),inΩ,u=0,on∂Ω,(P) 其中非线性 f 满足 Ambrosetti-Rabinowitz 条件。利用 f 在有界区域的附加增长条件，我们可以通过应用同调联系论证和莫尔斯理论得到 (P) 的五个非微观解。
{"title":"Multiple solutions of the Ambrosetti–Rabinowitz problem","authors":"Ziliang Yang , Jiabao Su , Mingzheng Sun","doi":"10.1016/j.aml.2024.109390","DOIUrl":"10.1016/j.aml.2024.109390","url":null,"abstract":"<div><div>In this paper, we consider the following elliptic problem <span><math><mrow><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace><mspace></mspace><mi>∂</mi><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mfenced><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> where the nonlinearity <span><math><mi>f</mi></math></span> satisfies the Ambrosetti–Rabinowitz condition. Using an additional growth condition of <span><math><mi>f</mi></math></span> at a bounded region, we can obtain five nontrivial solutions of <span><math><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></math></span> by applying homological linking arguments and Morse theory.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109390"},"PeriodicalIF":2.9,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-23DOI: 10.1016/j.aml.2024.109392
Yupeng Qin , Zhen Wang , Li Zou
The present work concerns with the higher dimensional Rayleigh–Plesset equation for describing the nonlinear dynamics of gas-filled hyper-spherical bubbles in an ideal fluid. A strict qualitative analysis is made by means of the bifurcation theory of dynamic system, indicating that the bubble oscillation type is periodic. An analytical approach based on elliptic function is suggested to construct parametric analytical solution with arbitrary space dimension , polytropic exponent and surface tension to the normalized higher dimensional Rayleigh–Plesset equation. The new obtained analytical solution extends the known ones for arbitrary (or some special cases of) and without considering the effect of surface tension. In addition, we also discuss the dynamic characteristics for the oscillating hyper-spherical bubbles.
{"title":"Qualitative analysis and analytical solution for higher dimensional gas-filled hyper-spherical bubbles in an ideal fluid","authors":"Yupeng Qin , Zhen Wang , Li Zou","doi":"10.1016/j.aml.2024.109392","DOIUrl":"10.1016/j.aml.2024.109392","url":null,"abstract":"<div><div>The present work concerns with the higher dimensional Rayleigh–Plesset equation for describing the nonlinear dynamics of gas-filled hyper-spherical bubbles in an ideal fluid. A strict qualitative analysis is made by means of the bifurcation theory of dynamic system, indicating that the bubble oscillation type is periodic. An analytical approach based on elliptic function is suggested to construct parametric analytical solution with arbitrary space dimension <span><math><mi>N</mi></math></span>, polytropic exponent <span><math><mi>κ</mi></math></span> and surface tension <span><math><mi>σ</mi></math></span> to the normalized higher dimensional Rayleigh–Plesset equation. The new obtained analytical solution extends the known ones for arbitrary (or some special cases of) <span><math><mi>N</mi></math></span> and <span><math><mi>κ</mi></math></span> without considering the effect of surface tension. In addition, we also discuss the dynamic characteristics for the oscillating hyper-spherical bubbles.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"162 ","pages":"Article 109392"},"PeriodicalIF":2.9,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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