Journal of Differential Equations最新文献

{"title":"Entire solutions to a quasilinear purely critical competitive system","authors":"Mónica Clapp,&nbsp;Víctor A. Vicente-Benítez","doi":"10.1016/j.jde.2025.113717","DOIUrl":"10.1016/j.jde.2025.113717","url":null,"abstract":"<div><div>We establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the <em>p</em>-Laplacian in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> whose nonlinear terms are purely critical.</div><div>We also show that the purely critical equation for the <em>p</em>-Laplacian in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> has infinitely many nodal solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113717"},"PeriodicalIF":2.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895057","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":"Metastability in parabolic equations and diffusion processes with a small parameter","authors":"M. Freidlin,&nbsp;L. Koralov","doi":"10.1016/j.jde.2025.113705","DOIUrl":"10.1016/j.jde.2025.113705","url":null,"abstract":"<div><div>We study diffusion processes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> that leave invariant a finite collection of manifolds (surfaces or points) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and small perturbations of such processes. Assuming certain ergodic properties at and near the invariant surfaces, we describe the rate at which the process gets attracted to or repelled from the surface, based on the local behavior of the coefficients. For processes that include, additionally, a small non-degenerate perturbation, we describe the metastable behavior. Namely, by allowing the time scale to depend on the size of the perturbation, we observe different asymptotic distributions of the process at different time scales.</div><div>Stated in PDE terms, the results provide the asymptotics, at different time scales, for the solution of the parabolic Cauchy problem when the operator that degenerates on a collection of surfaces is perturbed by a small non-degenerate term. This asymptotic behavior switches at a finite number of time scales that are calculated and does not depend on the perturbation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113705"},"PeriodicalIF":2.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895056","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":"Local well-posedness in Gevrey function spaces for 3D Boussinesq boundary layer system","authors":"Qian Li ,&nbsp;Peixin Wang ,&nbsp;Xiaojing Xu","doi":"10.1016/j.jde.2025.113725","DOIUrl":"10.1016/j.jde.2025.113725","url":null,"abstract":"<div><div>In this paper, we consider the 3D Boussinesq boundary layer system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, which is a coupling of the Prandtl type equations and a thermal layer equation due to the coupling of velocity and temperature in Boussinesq equations. We observe that there is also a cancellation mechanism in the temperature equation, which has been applied to the Prandtl equations in Li et al. (2022) <span><span>[14]</span></span>. Utilizing these cancellation mechanisms and constructing good unknowns, we overcome the loss of derivative arising in not only the velocity equations but also the temperature equation, then we show the local well-posedness of the Boussinesq boundary layer system in Gevrey function spaces. Furthermore, we obtain the optimal Gevrey index 2.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113725"},"PeriodicalIF":2.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895058","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":"Front propagation and blocking of time periodic bistable Lotka-Volterra competition-diffusion systems in cylindrical domains","authors":"Wei-Jie Sheng,&nbsp;Si-Jie Zang","doi":"10.1016/j.jde.2025.113718","DOIUrl":"10.1016/j.jde.2025.113718","url":null,"abstract":"<div><div>This paper is concerned with the propagation and blocking phenomena of time periodic bistable Lotka-Volterra competition-diffusion system in cylindrical domains. Firstly, we establish the existence of an entire solution emanating from a time periodic planar traveling front. Here the entire solution refers to a solution that is defined for all time and over the whole domain. Then we prove that the entire solution eventually converges to the same planar traveling front under the complete propagation condition when the domain is bilaterally straight. Finally, we give some sufficient conditions on the domain to ensure that the propagation of the entire solution is complete or be blocked.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113718"},"PeriodicalIF":2.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902746","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":"Nekhoroshev stability for random generalized Hamiltonian systems with different regularities","authors":"Bingqi Yu ,&nbsp;Yong Li","doi":"10.1016/j.jde.2025.113709","DOIUrl":"10.1016/j.jde.2025.113709","url":null,"abstract":"<div><div>In this article, we establish the Nekhoroshev stability of nearly integrable generalized Hamiltonian systems with bounded random perturbations possessing different regularity conditions. We generalize the original framework for proving the Nekhoroshev theorem. Using this unified framework, we can derive different normal form lemmas based on various regularity conditions, leading to results for stability times of different scales. Furthermore, this method allows perturbation functions with a certain degree of randomness and can be applied within the context of generalized Hamiltonian systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113709"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893821","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":"Hölder regularity for nonlocal in time subdiffusion equations with general kernel","authors":"Adam Kubica ,&nbsp;Katarzyna Ryszewska ,&nbsp;Rico Zacher","doi":"10.1016/j.jde.2025.113716","DOIUrl":"10.1016/j.jde.2025.113716","url":null,"abstract":"<div><div>We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work <span><span>[11]</span></span> by leaving the framework of distributed order fractional time derivatives and considering a general <span><math><mi>PC</mi></math></span> kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113716"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893062","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":"Heteroclinic bifurcation near a loop tangent to an invariant line","authors":"Xianbo Sun ,&nbsp;Guilin Ji ,&nbsp;Qun Bin","doi":"10.1016/j.jde.2025.113713","DOIUrl":"10.1016/j.jde.2025.113713","url":null,"abstract":"<div><div>In this paper, we propose a method for examining the heteroclinic bifurcation near a loop tangent to an invariant line in near-Hamiltonian systems. Our objective is to derive the asymptotic expansion of a generalized Melnikov function, which encompasses not only the first-order Melnikov function but also higher-order Melnikov functions in a wider range of reversible Hamiltonian systems. We apply our findings to a cubic reversible Hamiltonian system with polynomial perturbations of degree <em>n</em>. Our contributions include:</div><div><strong>(i)</strong> Determining the precise number of limit cycles near the tangent loop by using the first-order Melnikov function for polynomial perturbations of arbitrary degree <em>n</em>.</div><div><strong>(ii)</strong> Deriving all-order Melnikov functions with simplified expressions and integrable conditions for the system under the cubic perturbation (<span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>). Our analysis reveals that the first, second, third, and fourth-order Melnikov functions lead to the bifurcation of <span><math><mi>t</mi><mi>h</mi><mi>r</mi><mi>e</mi><mi>e</mi></math></span>, <span><math><mi>f</mi><mi>i</mi><mi>v</mi><mi>e</mi></math></span>, <em>six</em> limit cycles, and <em>one</em> limit cycle near the loop, respectively.</div><div><strong>(iii)</strong> Determining the exact upper bound on the maximum number of zeros of the first-order Melnikov function for the cubic perturbation by applying a modified Chebyshev criterion and an element-combination technique.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113713"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893717","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":"Averaging principle for SDEs with singular drifts driven by α-stable processes","authors":"Mengyu Cheng ,&nbsp;Zimo Hao ,&nbsp;Xicheng Zhang","doi":"10.1016/j.jde.2025.113706","DOIUrl":"10.1016/j.jde.2025.113706","url":null,"abstract":"<div><div>In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with <em>β</em>-Hölder drift driven by <em>α</em>-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>×</mo><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions, we obtain the optimal rate of strong convergence when <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo><mo>×</mo><mo>(</mo><mn>2</mn><mo>−</mo><mfrac><mrow><mn>3</mn><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>×</mo><mo>(</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. Furthermore, when <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>×</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>×</mo><mo>(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math></span>, we show the convergence of the martingale solutions of original systems to that of the averaged equation. In particular, when <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, the drift can be a distribution.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113706"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893718","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":"Almost sharp global wellposedness and scattering for the defocusing conformal wave equation on the hyperbolic space","authors":"Chutian Ma","doi":"10.1016/j.jde.2025.113714","DOIUrl":"10.1016/j.jde.2025.113714","url":null,"abstract":"<div><div>In this paper we prove a global well-posedness and scattering result for the defocusing conformal nonlinear wave equation in the hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>. We take advantage of the hyperbolic geometry which yields stronger Morawetz and Strichartz estimates. We show that the solution is globally wellposed and scatters if the initial data is radially symmetric and lies in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>, <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113714"},"PeriodicalIF":2.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885364","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":"On exact controllability for a class of 2-D Grushin hyperbolic equation","authors":"Donghui Yang ,&nbsp;Weijia Wu ,&nbsp;Bao-Zhu Guo ,&nbsp;Shugen Chai","doi":"10.1016/j.jde.2025.113710","DOIUrl":"10.1016/j.jde.2025.113710","url":null,"abstract":"<div><div>In this paper, we establish exact controllability for a class of two-dimensional Grushin hyperbolic equations, which are degenerate hyperbolic equations. Currently, research on degenerate equations primarily focuses on parabolic equations, with limited research on degenerate hyperbolic equations, mainly confined to one-dimensional cases. The primary method utilized in this paper for the hyperbolic Gurshin equation is through weighted Carleman estimates. Due to the degeneracy, classical Carleman estimates and regular solution spaces are not applicable to equations with degenerate coefficients. Therefore, we first introduce a weighted solution space. To prove controllability, we must first establish the existence of solutions, which is accomplished by adopting the standard Galerkin method. Once the existence of solutions is confirmed, we proceed to prove controllability by performing weighted Carleman estimates. Before calculations, we configure the weight function by adding a cut-off function to a solution of a similar but not dual equation. This enables us to truncate the degenerate part during calculations, turning it into zero, and eliminating the need to consider issues such as the regularity of the degenerate boundary and the meaningfulness of integrals on the degenerate boundary. Since we are integrating only in the non-degenerate region, we can proceed with Carleman estimates for regular non-degenerate hyperbolic equations. After obtaining the Carleman estimates, we use standard methods to prove the observability inequality, leading to the exact controllability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113710"},"PeriodicalIF":2.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885363","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}