Applied Mathematics Letters最新文献

{"title":"Global stability of reaction–diffusion equation with nonlocal delay","authors":"HuanHuan Qiu, Beijia Ren, Rong Zou","doi":"10.1016/j.aml.2024.109412","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109412","url":null,"abstract":"In this paper, we establish the global stability of the spatially nonhomogeneous steady state solution of a reaction diffusion equation with nonlocal delay under the Dirichlet boundary condition. To achieve this, we obtain the global existence and nonnegativity of solutions and give an extensive study on the properties of omega limit sets.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"6 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823228","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}

{"title":"Global dynamical behavior of a cholera model with temporary immunity","authors":"Ning Bai, Rui Xu","doi":"10.1016/j.aml.2024.109413","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109413","url":null,"abstract":"Existing studies have shown that asymptomatic cases might be related to short-term immunity on a timescale of weeks to months, which could have a significant impact on cholera epidemic transmission. In this paper, we are concerned with the global dynamical behavior of a cholera model with temporary immunity, which is characterized by discrete delay. The basic reproduction number of the model and the existence of each of feasible equilibria are studied. By using an iteration technique and comparison argument, sufficient conditions are obtained for the global attractivity of the endemic equilibrium.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"22 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823230","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}

{"title":"A study in Alzheimer’s disease model for pathological effect of oligomers on the interplay between [formula omitted]-amyloid and Ca2＋","authors":"Mingyan Dong, Yongxin Zhang, Gui-Quan Sun, Zun-Guang Guo, Jiao Zhang","doi":"10.1016/j.aml.2024.109407","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109407","url":null,"abstract":"Alzheimer’s disease (AD) is characterized by the progressive deposition of <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math>-amyloid (A<mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math>) plaques in the brain, where the A<mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math> oligomers have been confirmed to produce the critical cytotoxicity during the disease process. In this study, a model is established to describe the effect of A<mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math> oligomers on the interplay between A<mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math> and Ca<ce:sup loc=\"post\">2＋</ce:sup>. Mathematical analysis demonstrates the existence and stability of the equilibria and the conditions under which backward bifurcation and saddle–node bifurcation occur are proposed. In addition, the aggregate reproduction number <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> is introduced to characterize the progression of AD. These results may offer valuable insights for studying AD-related medical strategies.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"54 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790149","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}

{"title":"New asymptotic study on the non-autonomous NFDEs involving Haddock conjecture","authors":"Qian Wang","doi":"10.1016/j.aml.2024.109410","DOIUrl":"https://doi.org/10.1016/j.aml.2024.109410","url":null,"abstract":"The classical Haddock conjecture is extended to a kind of non-autonomous neutral functional differential equations (NFDEs) incorporating time-varying delays in this paper. By using the Dini derivative theory and inequality analyses, without requiring the strictly monotonically increasing property of the delay feedback function, it is demonstrated that every solution of the considered NFDEs is bounded and converges to a constant, which fully refines and generalizes the existing findings.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"33 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142790148","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}

{"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}

{"title":"Nonlocal ∂̄ formalism for the three-spatial-dimensions Kaup–Kuperschmidt equation with two temporal variables","authors":"Huanhuan Lu ,&nbsp;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}

{"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}

{"title":"Legendre spectral volume methods for Allen–Cahn equations by the direct discontinuous Galerkin formula","authors":"Chaoyue Guan,&nbsp;Yuli Sun,&nbsp;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}

{"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}

{"title":"Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations","authors":"Wei Zhang ,&nbsp;Jialing Zhang","doi":"10.1016/j.aml.2024.109402","DOIUrl":"10.1016/j.aml.2024.109402","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider this equation &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mfenced&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/mfenced&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Here &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denotes the &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;th elementary symmetric function of the eigenvalues of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;∇&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the gradient of &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denotes the Hessian of &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. 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}