Journal of Differential Equations最新文献

{"title":"Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori","authors":"","doi":"10.1016/j.jde.2024.09.044","DOIUrl":"10.1016/j.jde.2024.09.044","url":null,"abstract":"<div><div>In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (2014) <span><span>[11]</span></span> for the square case, to obtain the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>-instability (<span><math><mi>s</mi><mo>&gt;</mo><mn>1</mn></math></span>) of the elliptic equilibrium <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>. We also provide the existence of solutions <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> with arbitrarily small <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm which achieve a prescribed growth, say <span><math><msub><mrow><mo>‖</mo><mi>u</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub><mo>≥</mo><mi>K</mi><msub><mrow><mo>‖</mo><mi>u</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span>, <span><math><mi>K</mi><mo>≫</mo><mn>1</mn></math></span>, within a time <em>T</em> satisfying polynomial estimates, namely <span><math><mn>0</mn><mo>&lt;</mo><mi>T</mi><mo>&lt;</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> for some <span><math><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326436","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 central limit theorems for integrable Hamiltonian systems perturbed by white noise","authors":"","doi":"10.1016/j.jde.2024.09.047","DOIUrl":"10.1016/j.jde.2024.09.047","url":null,"abstract":"<div><div>In this paper, we consider the dynamics of integrable stochastic Hamiltonian systems. Utilizing the Nagaev-Guivarc'h method, we obtain several generalized results of the central limit theorem. Making use of this technique and the Birkhoff ergodic theorem, we prove that the invariant tori persist under stochastic perturbations. Moreover, they asymptotically follow a Gaussian distribution, which gives a positive answer to the stability of integrable stochastic Hamiltonian systems over time. Our results hold true for both Gaussian and non-Gaussian noises, and their intensities can be not small.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326437","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 the Borel summability of formal solutions of certain higher-order linear ordinary differential equations","authors":"","doi":"10.1016/j.jde.2024.09.041","DOIUrl":"10.1016/j.jde.2024.09.041","url":null,"abstract":"<div><div>We consider a class of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span>-order linear ordinary differential equations with a large parameter <em>u</em>. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of <em>u</em>. We demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter <em>u</em> in large, unbounded domains of the independent variable. We establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, we show that the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an <span><math><msup><mrow><mi>n</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span>-order Airy-type equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142323863","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":"Boundedness for the chemotaxis system with logistic growth","authors":"","doi":"10.1016/j.jde.2024.09.040","DOIUrl":"10.1016/j.jde.2024.09.040","url":null,"abstract":"<div><div>In this paper, we consider a mathematical model motivated by the studies of coral broadcast spawning<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>n</mi><mo>+</mo><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>n</mi><mo>−</mo><mi>Δ</mi><mi>n</mi></mtd><mtd><mo>=</mo><mo>−</mo><mi>χ</mi><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>∇</mi><mi>c</mi><mo>)</mo><mo>+</mo><mi>n</mi><mo>−</mo><mi>ϵ</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>c</mi><mo>+</mo><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>c</mi><mo>−</mo><mi>Δ</mi><mi>c</mi></mtd><mtd><mo>=</mo><mo>−</mo><mi>c</mi><mo>+</mo><mi>n</mi></mtd></mtr></mtable></mrow><mspace></mspace><mtext> in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>, <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>, and <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>. We establish global-in-time well-posedness and boundedness of the solution to the Cauchy problem of this system by developing local-in-space estimates. The crux point of our proof depends intensely on localization in the space of solutions induced by “local effect” of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>-norm.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320378","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":"Spectral instability of peakons for the b-family of Novikov equations","authors":"","doi":"10.1016/j.jde.2024.09.031","DOIUrl":"10.1016/j.jde.2024.09.031","url":null,"abstract":"<div><div>In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter <em>b</em>. This family is called the “<em>b</em>-Novikov” since it reduces to the integrable Novikov equation in the case <span><math><mi>b</mi><mo>=</mo><mn>3</mn></math></span>. By extending the corresponding linearized operator defined on functions in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> to one defined on weaker functions on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we prove spectral and linear instability on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of peakons in the <em>b</em>-Novikov equations for any <em>b</em>. We also consider the stability on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> and show that the peakons are spectrally or linearly stable only in the case <span><math><mi>b</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320377","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":"An Lq(Lp)-regularity theory for parabolic equations with integro-differential operators having low intensity kernels","authors":"","doi":"10.1016/j.jde.2024.09.033","DOIUrl":"10.1016/j.jde.2024.09.033","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&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;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&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;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;t&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;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; spaces. Our spatial operator &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;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; is an integro-differential operator of the form&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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;y&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;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; Here, &lt;span&gt;&lt;math&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; which yield &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&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;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-regularity of solutions. Our assumptions on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are general so that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; may be comparable to &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;ℓ&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;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a function &lt;em&gt;ℓ&lt;/em&gt; which is slowly varying at infinity. For example, we can take &lt;span&gt;&lt;math&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&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;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&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;α&lt;","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315770","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":"Limiting behavior of invariant foliations for SPDEs in singularly perturbed spaces","authors":"","doi":"10.1016/j.jde.2024.09.039","DOIUrl":"10.1016/j.jde.2024.09.039","url":null,"abstract":"<div><div>In this paper, we investigate a class of stochastic semilinear parabolic equations subjected to multiplicative noise within singularly perturbed phase spaces. We first establish the existence and smoothness of stable foliations. Then we prove that the long-term behavior of each solution is determined by a solution residing on the pseudo-unstable manifold via a leaf of the stable foliation. Finally, we present the convergence of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> invariant foliations as the high dimensional region collapse to low dimensional region. In contrast to the convergence of pseudo-unstable manifolds, we introduce a novel technique to address challenges arising from the singularity of the stable term of hyperbolicity in the proof of convergence of stable manifolds and stable foliations as the space collapses.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315771","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":"Exponential stability of a diffuse interface model of incompressible two-phase flow with phase variable dependent viscosity and vacuum","authors":"","doi":"10.1016/j.jde.2024.09.036","DOIUrl":"10.1016/j.jde.2024.09.036","url":null,"abstract":"<div><div>This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314634","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 the periodic solutions of switching scalar dynamical systems","authors":"","doi":"10.1016/j.jde.2024.09.032","DOIUrl":"10.1016/j.jde.2024.09.032","url":null,"abstract":"<div><div>In this paper, we investigate the existence and stability of periodic solutions of switching dynamical systems consisting of two sub-equations. We first establish a general criterion to determine the stability of periodic solutions; namely, we derive the conditions under which the periodic solution is locally asymptotically stable, globally asymptotically stable, or unstable. Next, we develop general theorems to count the number of periodic solutions and find the basins of attractions for the periodic solutions and the trivial solution, respectively. As applications, we analyze two biological models in recent literature. Our general theorems not only reproduce the existing results in a unified and simpler manner but also lead to new and complete dynamical results including bistability of the periodic solution and the trivial solution. Numerical examples are also given to illustrate our theoretical results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315767","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":"Maslov-type (L,P)-index and subharmonic P-symmetric brake solutions for Hamiltonian systems","authors":"","doi":"10.1016/j.jde.2024.09.037","DOIUrl":"10.1016/j.jde.2024.09.037","url":null,"abstract":"<div><div>This paper introduces a novel iteration inequality for the Maslov-type <span><math><mo>(</mo><mi>L</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>-index of iterated symplectic paths. Here, <em>P</em> is a fixed 2<em>n</em>-dimensional symplectic and orthogonal matrix satisfying <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><mi>I</mi></math></span>. These advancements in index theory are then applied to investigate the multiplicity of subharmonic solutions in Hamiltonian systems exhibiting dihedral equivariance with period <em>mτ</em>. Notably, a criterion of geometric distinction is established for two subharmonic <em>P</em>-symmetric brake orbits with periods <em>kmτ</em> and <em>lmτ</em> within the set <span><math><mo>{</mo><mi>k</mi><mi>m</mi><mi>τ</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>k</mi><mo>≡</mo><mn>1</mn><mtext> (mod </mtext><mi>m</mi><mo>)</mo><mo>}</mo></math></span>. This criterion is based on a lower bound estimate for the ratio <span><math><mi>l</mi><mo>/</mo><mi>k</mi></math></span>. Specifically, for odd <em>k</em>, the lower bound must be not less than <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>dim</mi><mo>⁡</mo><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>P</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mi>m</mi><mo>+</mo><mn>1</mn></math></span>, while for even <em>k</em>, it must be not less than <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>dim</mi><mo>⁡</mo><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>P</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mi>m</mi><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315768","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}