Journal of Differential Equations最新文献

{"title":"On uniqueness of KP soliton structures","authors":"Francisco Alegría ,&nbsp;Gong Chen ,&nbsp;Claudio Muñoz ,&nbsp;Felipe Poblete ,&nbsp;Benjamín Tardy","doi":"10.1016/j.jde.2025.113569","DOIUrl":"10.1016/j.jde.2025.113569","url":null,"abstract":"<div><div>We consider the Kadomtsev-Petviashvili II (KP) model placed in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>×</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>, in the case of smooth data that are not necessarily in a Sobolev space. In this paper, the subclass of smooth solutions we study is of “soliton type”, characterized by a phase <span><math><mi>Θ</mi><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> and a unidimensional profile <em>F</em>. In particular, every classical KP soliton and multi-soliton falls into this category with suitable Θ and <em>F</em>. We establish concrete characterizations of KP solitons by means of a natural set of nonlinear differential equations and inclusions of functionals of Wronskian, Airy and Heat types, among others. These functional equations only depend on the new variables Θ and <em>F</em>. A distinct characteristic of this set of functionals is its special and rigid structure tailored to the considered soliton. By analyzing Θ and <em>F</em>, we establish the uniqueness of line-solitons, multi-solitons, and other degenerate solutions among a large class of KP solutions. Our results are also valid for other 2D dispersive models such as the quadratic and cubic Zakharov-Kuznetsov equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113569"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490404","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":"Asymptotic behavior of small solutions to a 2-component system of cubic nonlinear Schrödinger equations in one space dimension","authors":"Yuji Sagawa","doi":"10.1016/j.jde.2025.113576","DOIUrl":"10.1016/j.jde.2025.113576","url":null,"abstract":"<div><div>In this manuscript we specify asymptotic behavior of small solutions to initial value problem for a 2-component system of cubic nonlinear Schrödinger equations in one dimensional Euclidean space. As a consequence, the solution behaves like a free solution as <span><math><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span>. Moreover, a non-decay result for the solution is derived, which is non-trivial in terms of the long range scattering.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113576"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490448","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":"Determining Lorentzian manifold from non-linear wave observation at a single point","authors":"Medet Nursultanov ,&nbsp;Lauri Oksanen ,&nbsp;Leo Tzou","doi":"10.1016/j.jde.2025.113563","DOIUrl":"10.1016/j.jde.2025.113563","url":null,"abstract":"<div><div>We consider an inverse problem for a non-linear hyperbolic equation. We show that the conformal structure of a Lorentzian manifold can be determined by the source-to-solution map evaluated along a single timelike curve. We use the microlocal analysis of non-linear wave interaction.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113563"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490506","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":"Stability and blow-up result for a class of a generalized Klein-Gordon equation","authors":"Claudianor O. Alves ,&nbsp;Paulo Cesar Carrião ,&nbsp;André Vicente","doi":"10.1016/j.jde.2025.113590","DOIUrl":"10.1016/j.jde.2025.113590","url":null,"abstract":"<div><div>In this paper we prove the existence of solution to a generalized Klein-Gordon equation with damping and source terms. The space derivative part of the main operator is described by a pseudo-differential operator given by <span><math><mo>−</mo><mi>Δ</mi><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mo>⋅</mo><mo>)</mo></math></span>, where Δ is the Euclidean Laplace operator and <em>c</em> is a positive constant. To prove the existence solution we introduced an appropriate structure of Hilbert spaces which allows us to use semigroups theory when the damping term is nonlinear. Using the Nehari manifold associated to the stationary problem, we create a stable set <span><math><mi>S</mi></math></span> such that, taking the initial data in <span><math><mi>S</mi></math></span>, the solution is global and the energy of the problem decay exponentially. In this case the damping is nonlinear and the source term satisfies the general assumption known as Ambrosetti-Rabinowitz condition. Moreover, under some appropriate conditions on the initial data we also prove a blow-up result with the source term subject to the Ambrosetti-Rabinowitz condition. Finally, we also prove a stability result with a more restrictive source term, which allows characterize the pass mountain level of the stationary problem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113590"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490351","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":"Solutions with prescribed mass for the p-Laplacian Schrödinger-Poisson system with critical growth","authors":"Kai Liu ,&nbsp;Xiaoming He ,&nbsp;Vicenţiu D. Rădulescu","doi":"10.1016/j.jde.2025.113570","DOIUrl":"10.1016/j.jde.2025.113570","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we focus on the existence and multiplicity of solutions for the &lt;em&gt;p&lt;/em&gt;-Laplacian Schrödinger-Poisson system&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&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;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&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;mi&gt;λ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&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;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&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;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&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;msup&gt;&lt;mrow&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&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;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&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;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; with a prescribed mass given by&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&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;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; in the Sobolev critical case, where, &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; are parameters, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is the Sobolev critical exponent, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is an undetermined parameter, acting as a Lagrange multiplier. We investigate this system under the &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-subcritical perturbation &lt;span&gt;&lt;math&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, with &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and establish the existence of multiple normalized solutions using the truncation technique, concentration-compactness ","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113570"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490447","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":"Heat-source type atmospheric nonlinear flow patterns in zonal cloud bands","authors":"C.I. Martin","doi":"10.1016/j.jde.2025.113589","DOIUrl":"10.1016/j.jde.2025.113589","url":null,"abstract":"<div><div>We present a family of exact solutions to a set of recently derived nonlinear equations governing at leading order the dynamics of flows in zonal cloud bands that resemble those on Jupiter. These solutions are radial in the horizontal variables, present density and temperature that decrease with height, a pressure function that decreases in the radial direction, and allow heat flowing out into the environment: these are features that are also observed in the Jupiter's Red Spot. Using a WKB analysis we show that certain exact solutions are stable under a specific choice of the density distribution.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113589"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490405","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":"Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity","authors":"Wenhui Chen ,&nbsp;Giovanni Girardi","doi":"10.1016/j.jde.2025.113568","DOIUrl":"10.1016/j.jde.2025.113568","url":null,"abstract":"<div><div>In this paper we consider semilinear wave equation and semilinear second order <em>σ</em>-evolution equations with different (effective or non-effective) damping mechanisms driven by fractional Laplace operators; in particular, the nonlinear term is the product of a power nonlinearity <span><math><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup></math></span> with the critical exponent <span><math><mi>p</mi><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and a modulus of continuity <span><math><mi>μ</mi><mo>(</mo><mo>|</mo><mi>u</mi><mo>|</mo><mo>)</mo></math></span>. We derive a critical condition on the nonlinearity by proving a global in time existence result under the Dini condition on <em>μ</em> and a blow-up result when <em>μ</em> does not satisfy the Dini condition. Especially, in this latter case we determine new sharp estimates for the lifespan of local solutions, obtaining coincident upper and lower bounds of the lifespan. In particular, we derive a new sharp estimate for the wave equation with structural damping and classical power nonlinearity <span><math><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup></math></span> in the critical case <span><math><mi>p</mi><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, not yet determined in previous literature. The proof of the blow-up results and the upper bound estimates of the lifespan require the introduction of new test functions which allows to overcome some new difficulties due to the presence of both non-local differential operators and general nonlinearities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113568"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491147","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":"Stability and large-time behavior for Euler-like equations","authors":"Jiahong Wu ,&nbsp;Xiaojing Xu ,&nbsp;Yueyuan Zhong ,&nbsp;Ning Zhu","doi":"10.1016/j.jde.2025.113578","DOIUrl":"10.1016/j.jde.2025.113578","url":null,"abstract":"<div><div>This paper intends to understand the long-time existence and stability of solutions to an Euler-like equation. An Euler-like equation is the 2D incompressible Euler equation with an extra singular integral operator (SIO) type term. In contrast to the 2D Euler equation, the vorticity to the 2D Euler-like equation is not known to be bounded due to the unboundedness of the SIO on the space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>. As a consequence, classical Yudovich theory fails on the Euler-like equation. The global existence, regularity and stability problems on the Euler-like equation are generally open. This paper makes progress on an Euler-like equation arising in the study of several fluids. We establish a long-time existence and stability result. When the Sobolev size of the initial data is of order <em>ε</em>, the solution is shown to live on a time interval of the size <span><math><mn>1</mn><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. When the initial data is restricted to a class with special symmetry, we obtain the global existence and nonlinear stability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113578"},"PeriodicalIF":2.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491342","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":"Regularity of the 3D stochastic viscous Primitive Equations","authors":"Zhao Dong ,&nbsp;Hao Xiong ,&nbsp;Guoli Zhou","doi":"10.1016/j.jde.2025.113579","DOIUrl":"10.1016/j.jde.2025.113579","url":null,"abstract":"<div><div>Utilizing the method of hydrostatic decomposition, we obtain the smoothness property and uniform <em>a</em> <span><math><mi>p</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>r</mi><mi>i</mi></math></span> estimates for the strong solution to 3D stochastic Primitive Equations (PEs) of large-scale ocean and atmosphere dynamics with non-periodic condition. Consequently, we derive the existence of invariant measures and the smoothness of random attractor.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113579"},"PeriodicalIF":2.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144480526","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 stability of traveling waves for Nagumo equations with degenerate diffusion","authors":"Tianyuan Xu ,&nbsp;Shanming Ji ,&nbsp;Ming Mei ,&nbsp;Jingxue Yin","doi":"10.1016/j.jde.2025.113587","DOIUrl":"10.1016/j.jde.2025.113587","url":null,"abstract":"<div><div>This paper is concerned with the global nonlinear stability with possibly large perturbations of the unique sharp / smooth traveling waves for the degenerate diffusion equations with Nagumo (bistable) reaction. Two technical issues arise in this study. One is the shortage of weak regularity of sharp traveling waves, the other difficulty is the non-absorbing initial-perturbation around the smooth traveling waves at the far field <span><math><mi>x</mi><mo>=</mo><mo>+</mo><mo>∞</mo></math></span>. For the sharp traveling wave case, we technically construct weak sub- and super-solutions with semi-compact supports via translation and scaling of the unique sharp traveling wave to characterize the motion of the steep moving edges and avoid the weak regularity of the solution near the steep edges. For the smooth traveling wave case, we artfully combine both the translation and scaling type sub- and super-solutions and the translation and superposition type sub- and super-solutions in a systematical manner.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113587"},"PeriodicalIF":2.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144490352","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}