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Universal gradient estimates of Δu + a(x)up(ln⁡(u+c))q = 0 on complete Riemannian manifolds
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-24 DOI: 10.1016/j.jde.2025.113257
Chong Song, Jibo Wu
{"title":"Universal gradient estimates of Δu + a(x)up(ln⁡(u+c))q = 0 on complete Riemannian manifolds","authors":"Chong Song,&nbsp;Jibo Wu","doi":"10.1016/j.jde.2025.113257","DOIUrl":"10.1016/j.jde.2025.113257","url":null,"abstract":"<div><div>In this paper, we study the elliptic non-linear equation <span><math><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><msup><mrow><mo>(</mo><mi>ln</mi><mo>⁡</mo><mo>(</mo><mi>u</mi><mo>+</mo><mi>c</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> on a complete Riemannian manifold with Ricci curvature bounded from below. By applying Nash-Moser iteration, we establish universal gradient estimates for positive solutions to the equation, where <span><math><mi>c</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is allowed to change sign. As an application, we obtain Liouville theorems when the manifold has non-negative Ricci curvature and <span><math><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is constant.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113257"},"PeriodicalIF":2.4,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684984","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}
引用次数: 0
Local well-posedness of the minimum energy estimator for a defocusing cubic wave equation
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-22 DOI: 10.1016/j.jde.2025.113258
Jesper Schröder
{"title":"Local well-posedness of the minimum energy estimator for a defocusing cubic wave equation","authors":"Jesper Schröder","doi":"10.1016/j.jde.2025.113258","DOIUrl":"10.1016/j.jde.2025.113258","url":null,"abstract":"<div><div>This work is concerned with the <em>minimum energy estimator</em> for a nonlinear hyperbolic partial differential equation. The <em>Mortensen observer</em> – originally introduced for the energy-optimal reconstruction of the state of nonlinear finite-dimensional systems – is formulated for a disturbed cubic wave equation and the associated observer equation is derived. An in depth study of the associated optimal control problem and sensitivity analysis of the corresponding value function reveals that the energy optimal state estimator is well-defined. Deploying a classical fixed point argument we proceed to show that the observer equation is locally well-posed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113258"},"PeriodicalIF":2.4,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Backward stochastic Volterra integral equations with jumps and some related problems
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-21 DOI: 10.1016/j.jde.2025.113240
Zongkui Fu, Shasha Shen, Jinbiao Wu
{"title":"Backward stochastic Volterra integral equations with jumps and some related problems","authors":"Zongkui Fu,&nbsp;Shasha Shen,&nbsp;Jinbiao Wu","doi":"10.1016/j.jde.2025.113240","DOIUrl":"10.1016/j.jde.2025.113240","url":null,"abstract":"<div><div>In this paper, we deal with backward stochastic Volterra integral equations with jumps. Firstly, we present the well-posedness of backward stochastic Volterra integral equations with jumps in the sense of adapted M-solution. Secondly, we give some properties of backward stochastic Volterra integral equations with jumps, which contain the duality principle, comparison theorem and the regularity of adapted M-solution. Thirdly, dynamic risk measure by means of backward stochastic Volterra integral equations with jumps is established. Fourthly, a maximum principle of Pontryagin type is obtained for an optimal control problem of stochastic Volterra integral equations with jumps. Finally, we investigate the well-posedness of linear fractional backward stochastic Volterra integral equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113240"},"PeriodicalIF":2.4,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683569","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}
引用次数: 0
Two classes of Benjamin–Ono-type equations with the Hilbert operator related to the Calogero–Moser system and the classical orthogonal polynomials
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-21 DOI: 10.1016/j.jde.2025.113249
Ling-Juan Yan , Ya-Jie Liu , Xing-Biao Hu , Ying-Nan Zhang
{"title":"Two classes of Benjamin–Ono-type equations with the Hilbert operator related to the Calogero–Moser system and the classical orthogonal polynomials","authors":"Ling-Juan Yan ,&nbsp;Ya-Jie Liu ,&nbsp;Xing-Biao Hu ,&nbsp;Ying-Nan Zhang","doi":"10.1016/j.jde.2025.113249","DOIUrl":"10.1016/j.jde.2025.113249","url":null,"abstract":"<div><div>This paper investigates two distinct classes of Benjamin–Ono(BO)-type equations with the Hilbert operator. The first class consists of equations with constant coefficients, derived from linear differential equations, with a specific focus on the Mikhailov–Novikov equation and Satsuma–Mimura equation. The second class involves BO-type equations with variable coefficients linked to orthogonal polynomials, including Hermite, Jacobi, and Laguerre polynomials. A key aspect of transforming these differential equations into BO-type equations is that the zeros of the polynomial or periodic solutions must lie in the upper half-plane. For linear and quadratic polynomials, we directly analyze their zeros to determine the solutions of corresponding BO-type equations. For higher-order polynomials, we use the pole expansion method to derive the governing many-body systems of the zeros. This study deepens our understanding of the relationship between the zeros of polynomials and the solutions of BO-type equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113249"},"PeriodicalIF":2.4,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685427","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}
引用次数: 0
Local controllability of the Korteweg-de Vries equation with the right Dirichlet control
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113235
Hoai-Minh Nguyen
{"title":"Local controllability of the Korteweg-de Vries equation with the right Dirichlet control","authors":"Hoai-Minh Nguyen","doi":"10.1016/j.jde.2025.113235","DOIUrl":"10.1016/j.jde.2025.113235","url":null,"abstract":"<div><div>The Korteweg-de Vries (KdV) equation with the right Dirichlet control is small time, locally, exactly controllable for all non-critical lengths and its linearized system is not controllable for <em>all</em> critical lengths. In this paper, we give a definitive picture of the local controllability properties of this control problem for all critical lengths. In particular, we show that the unreachable space of the linearized system is always of dimension 1 and the KdV system with the right Dirichlet control is not locally null controllable in small time for any critical length. We also give a criterion to determine whether the system is locally exactly controllable in finite time or <em>not</em> locally null controllable in any positive time for <em>all</em> critical lengths. Consequently, we show that there exist critical lengths such that the system is <em>not</em> locally null controllable in small time but is locally exactly controllable in finite time.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113235"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683565","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}
引用次数: 0
Global existence of strong solutions to the Landau–Lifshitz–Slonczewski equation
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113242
Chenlu Zhang, Huaqiao Wang
{"title":"Global existence of strong solutions to the Landau–Lifshitz–Slonczewski equation","authors":"Chenlu Zhang,&nbsp;Huaqiao Wang","doi":"10.1016/j.jde.2025.113242","DOIUrl":"10.1016/j.jde.2025.113242","url":null,"abstract":"<div><div>In this paper, we focus on the existence of strong solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation. We construct a new combination of Bourgain space and Lebesgue space where linear and nonlinear estimates can be closed by applying frequency decomposition and energy methods. Finally, we establish the existence and uniqueness of the global strong solution provided that the initial data belongs to Besov space <span><math><msubsup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>Ω</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113242"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685426","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}
引用次数: 0
The Neumann problem for a class of Hessian quotient type equations
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113251
Jiabao Gong, Zixuan Liu, Qiang Tu
{"title":"The Neumann problem for a class of Hessian quotient type equations","authors":"Jiabao Gong,&nbsp;Zixuan Liu,&nbsp;Qiang Tu","doi":"10.1016/j.jde.2025.113251","DOIUrl":"10.1016/j.jde.2025.113251","url":null,"abstract":"<div><div>In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the <span><math><mo>(</mo><mi>Λ</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-convex solution of Hessian quotient equation <span><math><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>Λ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>Λ</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow></mfrac><mo>=</mo><mi>ψ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>l</mi><mo>&lt;</mo><mi>k</mi><mo>≤</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msubsup></math></span> under the assumption of the growth condition. As an application, we obtain the global a priori estimates and the existence theorem for the Neumann problem of this Hessian quotient type equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"431 ","pages":"Article 113251"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683312","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}
引用次数: 0
Global boundedness of solutions to a class of partial differential equations with time delay
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113232
Xuanyu Liu , Junping Shi , Chuncheng Wang , Dejun Fan
{"title":"Global boundedness of solutions to a class of partial differential equations with time delay","authors":"Xuanyu Liu ,&nbsp;Junping Shi ,&nbsp;Chuncheng Wang ,&nbsp;Dejun Fan","doi":"10.1016/j.jde.2025.113232","DOIUrl":"10.1016/j.jde.2025.113232","url":null,"abstract":"<div><div>A class of diffusive partial differential equations with strongly coupled time delays and diffusion is considered. The global boundedness of weak solutions of the equation is proved by an entropy method that was initially proposed for studying the global boundedness of reaction-diffusion equations with cross-diffusion. The presence of the time delays in the equation prevents the entropy method to be directly applied, and here we extend the entropy method to this class of diffusive partial differential equations with time delays by proving some key entropy inequalities, which further allows us to obtain the estimates of gradient of the solutions. The results can be used to show the global boundedness of solutions of population models with memory effect, which were recently proposed for describing the movement of highly-developed animal species. In addition, we show that the results are also applicable for the classic partial functional differential equations, where the time delays only appear in the reaction terms.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113232"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683567","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}
引用次数: 0
A simple way to well-posedness in H1 of a delay differential equation from cell biology
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113241
Bernhard Aigner , Marcus Waurick
{"title":"A simple way to well-posedness in H1 of a delay differential equation from cell biology","authors":"Bernhard Aigner ,&nbsp;Marcus Waurick","doi":"10.1016/j.jde.2025.113241","DOIUrl":"10.1016/j.jde.2025.113241","url":null,"abstract":"<div><div>We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations <span><span>[10]</span></span> to a generalized population model for stem cell maturation. The weak approach using Sobolev-spaces we take allows for a larger class of initial prehistories and makes checking the requirements for well-posedness of such a model considerably easier compared to previous approaches. In fact the present approach is a possible means to guarantee that the solution manifold is not empty, which is a necessary requirement for a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-approach to work.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113241"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sub-exponential localization for a random tight-binding model with long-range hopping
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-03-20 DOI: 10.1016/j.jde.2025.113239
Siqi Xu , Dongfeng Yan
{"title":"Sub-exponential localization for a random tight-binding model with long-range hopping","authors":"Siqi Xu ,&nbsp;Dongfeng Yan","doi":"10.1016/j.jde.2025.113239","DOIUrl":"10.1016/j.jde.2025.113239","url":null,"abstract":"<div><div>In this paper, we study the Anderson tight-binding model on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with the sub-exponential long-range hopping and log-Hölder continuously distributed potential. It is proved that, at high disorder this model has pure point spectrum with sub-exponentially decaying eigenfunctions. This gives a partial answer to a conjecture of Yeung-Oono [<em>Europhys. Lett.</em> 4(9), (1987): 1061-1065]. Our proof is based on multi-scale analysis type Green's function estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"431 ","pages":"Article 113239"},"PeriodicalIF":2.4,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683311","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}
引用次数: 0
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