Journal of Differential Equations最新文献

{"title":"Normalized solutions for a nonlinear Dirac equation","authors":"Vittorio Coti Zelati ,&nbsp;Margherita Nolasco","doi":"10.1016/j.jde.2024.09.029","DOIUrl":"10.1016/j.jde.2024.09.029","url":null,"abstract":"<div><div>We prove the existence of a normalized, stationary solution <span><math><mi>ψ</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> with frequency <span><math><mi>ω</mi><mo>&gt;</mo><mn>0</mn></math></span> of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form<span><span><span><math><mi>F</mi><mo>(</mo><mi>Ψ</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo>|</mo><mo>(</mo><mi>Ψ</mi><mo>,</mo><msup><mrow><mi>γ</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>Ψ</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mi>b</mi><mo>|</mo><mo>(</mo><mi>Ψ</mi><mo>,</mo><msup><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>γ</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>Ψ</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span></span></span> with <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>]</mo></math></span>, <span><math><mi>b</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span> sufficiently small. Here <span><math><msup><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msup></math></span>, <span><math><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>3</mn></math></span> are the <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> Dirac's matrices.</div><div>We find the solution as a critical point of a suitable functional restricted to the unit sphere in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and <em>ω</em> turns out to be the corresponding Lagrange multiplier.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022039624006144/pdfft?md5=cb690464016ef3752322a3f835e48f7c&pid=1-s2.0-S0022039624006144-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312201","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}

{"title":"The isochronal phase of stochastic PDE and integral equations: Metastability and other properties","authors":"Zachary P. Adams ,&nbsp;James MacLaurin","doi":"10.1016/j.jde.2024.09.002","DOIUrl":"10.1016/j.jde.2024.09.002","url":null,"abstract":"<div><div>We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighborhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T. Winfree and J. Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>, but less than <span><math><mi>O</mi><mo>(</mo><mi>exp</mi><mo>⁡</mo><mo>(</mo><mi>C</mi><msup><mrow><mi>σ</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>σ</mi><mo>≪</mo><mn>1</mn></math></span> is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312202","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":"The focusing complex mKdV equation with nonzero background: Large N-order asymptotics of multi-rational solitons and related Painlevé-III hierarchy","authors":"Weifang Weng ,&nbsp;Guoqiang Zhang ,&nbsp;Zhenya Yan","doi":"10.1016/j.jde.2024.09.038","DOIUrl":"10.1016/j.jde.2024.09.038","url":null,"abstract":"<div><div>In this paper, we investigate the large-order asymptotics of multi-rational solitons of the focusing complex modified Korteweg-de Vries (c-mKdV) equation with nonzero background via the Riemann-Hilbert problems. First, based on the Lax pair, inverse scattering transform, and a series of deformations, we construct a multi-rational soliton of the c-mKdV equation via a solvable Riemann-Hilbert problem (RHP). Then, through a scale transformation, we construct a RHP corresponding to the limit function which is a new solution of the c-mKdV equation in the rescaled variables <span><math><mi>X</mi><mo>,</mo><mspace></mspace><mi>T</mi></math></span>, and prove the existence and uniqueness of the RHP's solution. Moreover, we also find that the limit function satisfies the ordinary differential equations (ODEs) with respect to space <em>X</em> and time <em>T</em>, respectively. The ODEs with respect to space <em>X</em> are identified with certain members of the Painlevé-III hierarchy. We study the large <em>X</em> and transitional asymptotic behaviors of near-field limit solutions, and we provide some part results for the case of large <em>T</em>. These results will be useful to understand and apply the large-order rational solitons in the nonlinear wave equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142310595","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 for the fast diffusion equation with absorption and singularity","authors":"Changping Xie ,&nbsp;Shaomei Fang ,&nbsp;Ming Mei ,&nbsp;Yuming Qin","doi":"10.1016/j.jde.2024.09.026","DOIUrl":"10.1016/j.jde.2024.09.026","url":null,"abstract":"<div><p>This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>△</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies <span><math><mn>0</mn><mo>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span> and the absorption index is <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span>. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for <span><math><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span> and <span><math><mi>p</mi><mo>&gt;</mo><mi>m</mi><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span> via the entropy dissipation method combining the generalized Shannon's inequality and Csiszár-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272360","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":"Existence, uniqueness and interior regularity of viscosity solutions for a class of Monge-Ampère type equations","authors":"Mengni Li ,&nbsp;You Li","doi":"10.1016/j.jde.2024.09.024","DOIUrl":"10.1016/j.jde.2024.09.024","url":null,"abstract":"<div><p>The Monge-Ampère type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Ampère type equations, which can be degenerate or singular near the boundary of convex domains. Viscosity subsolutions and viscosity supersolutions to the problem can be constructed via comparison principle. Finally, we demonstrate the existence, uniqueness and a series of interior regularities (including <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi></mrow></msup></math></span> with <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span>, <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>μ</mi></mrow></msup></math></span> with <span><math><mi>μ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>) of the viscosity solution to the problem.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274259","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":"Improved blow-up criteria for some Camassa-Holm type equations","authors":"Rudong Zheng","doi":"10.1016/j.jde.2024.09.022","DOIUrl":"10.1016/j.jde.2024.09.022","url":null,"abstract":"<div><p>We study the blow-up phenomena for some integrable Camassa-Holm type equations on the line. For the two-component Camassa-Holm system, we give a sufficient condition on the initial data that leads to a blow-up. For the Degasperis-Procesi equation, we establish a local-in-space blow-up criterion which improves considerably the early criterion based on the sign-changing momentum. Besides, we obtain some new blow-up criteria for the Novikov equation and the modified Camassa-Holm equation.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274258","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":"Uniform-in-time stability and continuous transition of the time-discrete infinite Kuramoto model","authors":"Seung-Yeal Ha ,&nbsp;Eun Taek Lee ,&nbsp;Wook Yoon","doi":"10.1016/j.jde.2024.09.021","DOIUrl":"10.1016/j.jde.2024.09.021","url":null,"abstract":"<div><p>We study a continuous transition from the discrete infinite Kuramoto model to the continuous counterpart in a whole time interval. The discrete infinite Kuramoto model corresponds to the discretization of the infinite Kuramoto model <span><span>[18]</span></span> via the first-order Euler discretization algorithm. For the proposed discrete infinite Kuramoto model, we study the emergent dynamics and uniform (-in-time) stability with respect to initial data under a suitable framework which is formulated in terms of system parameters and initial data. For a homogeneous ensemble with the same natural frequencies, we identify sufficient conditions for the existence of “quasi-stationary state” and complete synchronization. In contrast, for a heterogeneous ensemble, we also provide a weak emergent dynamics, namely “practical synchronization”. For the continuous transition in a zero time-step limit, we provide an improved truncation error estimate compared to the error estimate which can be obtained from the general theory for first-order discretized model using the uniform stability and emergent dynamics.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274363","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 Hénon type problem with nonlinearities involving slightly subcritical growth","authors":"Imene Bendahou ,&nbsp;Zied Khemiri ,&nbsp;Fethi Mahmoudi","doi":"10.1016/j.jde.2024.09.016","DOIUrl":"10.1016/j.jde.2024.09.016","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem&lt;span&gt;&lt;span&gt;&lt;span&gt;(0.1)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&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;mi&gt;p&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in &lt;/mtext&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in &lt;/mtext&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&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;mi&gt;s&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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is the Sobolev critical exponent, &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is a smooth bounded domain with Lipschitz boundary, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is the fractional Laplace operator and &lt;span&gt;&lt;math&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a bounded positive continuous function. We assume that there exists a nondegenerate critical point &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ξ&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;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of the restriction of &lt;em&gt;β&lt;/em&gt; to the boundary ∂Ω such that&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ξ&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;/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;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ξ&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;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Given any integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we show that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; small enough, problem &lt;span&gt;&lt;span&gt;(0.1)&lt;/span&gt;&lt;/span&gt; has a positive solution, which is a sum of &lt;em&gt;k&lt;/em&gt; bubbles which concentrate at &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; as &lt;em&gt;ε&lt;/em&gt; tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. This work can be seen as a nonloca","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272359","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 stochastic mosquito population suppression model based on incomplete cytoplasmic incompatibility and time switching","authors":"Rong Yan ,&nbsp;Wenjuan Guo ,&nbsp;Jianshe Yu","doi":"10.1016/j.jde.2024.09.017","DOIUrl":"10.1016/j.jde.2024.09.017","url":null,"abstract":"<div><p>In this paper, we establish and study a stochastic mosquito population suppression model incorporating the release of <em>Wolbachia</em>-infected males and time switching, where stochastic noises are given by independent standard Brownian motions. By combining the actual mosquito control strategy in Guangzhou, we assume that the waiting release period <em>T</em> between two consecutive releases of <em>Wolbachia</em>-infected males is less than the sexually active lifespan <span><math><mover><mrow><mi>T</mi></mrow><mo>‾</mo></mover></math></span> of them. The existence and uniqueness of global positive solutions and stochastically ultimate boundedness for the stochastic model are obtained. Some sufficient conditions for the extinction and the existence of stochastic non-trivial periodic solutions are established. Furthermore, we assume that the release function is a general periodic function and some stochastic dynamical behaviors are obtained. Numerical examples are presented to illustrate the theoretical results.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274257","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":"Wasserstein convergence rate of invariant measures for stochastic Schrödinger delay lattice systems","authors":"Zhang Chen ,&nbsp;Dandan Yang ,&nbsp;Shitao Zhong","doi":"10.1016/j.jde.2024.08.065","DOIUrl":"10.1016/j.jde.2024.08.065","url":null,"abstract":"<div><p>This paper is concerned with the convergence of invariant measures in the Wasserstein sense for the stochastic Schrödinger delay lattice systems as delay parameter <em>ρ</em> or parameter <em>β</em> approaches zero. Through <em>p</em>th-order moment estimates of solutions to systems, as well as the Hölder continuity estimates of solutions with respect to time, we obtain the convergence of solutions about initial data and the above parameters. Then together with high-order moment estimates of invariant measures, we prove that the unique invariant measure of such delay lattice system converges to the invariant measure of limiting system in the Wasserstein sense as delay parameter <em>ρ</em> or parameter <em>β</em> approaches zero, and the corresponding convergence rate is also obtained.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142274255","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}