Journal of Differential Equations最新文献

{"title":"Compactness of Green operators with applications to semilinear nonlocal elliptic equations","authors":"Phuoc-Truong Huynh ,&nbsp;Phuoc-Tai Nguyen","doi":"10.1016/j.jde.2024.11.019","DOIUrl":"10.1016/j.jde.2024.11.019","url":null,"abstract":"<div><div>In this paper, we consider a class of integro-differential operators <span><math><mi>L</mi></math></span> posed on a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>Ω</mi></mrow></msup></math></span>. Under mild conditions on <span><math><mi>L</mi></math></span> and its Green operator, we establish various sharp compactness of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>Ω</mi></mrow></msup></math></span> involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation <span><math><mi>L</mi><mi>u</mi><mo>+</mo><mi>g</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>μ</mi></math></span> in Ω with boundary condition <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> on ∂Ω or exterior condition <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi></math></span> if applicable, where <em>μ</em> is a Radon measure on Ω and <span><math><mi>g</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> is a nondecreasing continuous function satisfying a subcriticality integral condition. When <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>t</mi></math></span> with <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span>, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 97-141"},"PeriodicalIF":2.4,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703309","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":"Quantitative profile decomposition and stability for a nonlocal Sobolev inequality","authors":"Paolo Piccione ,&nbsp;Minbo Yang ,&nbsp;Shunneng Zhao","doi":"10.1016/j.jde.2024.11.013","DOIUrl":"10.1016/j.jde.2024.11.013","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given by&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&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;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;⁎&lt;/mo&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;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&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;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&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;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&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;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&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;mn&gt;2&lt;/mn&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;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;∀&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where ⁎ denotes the convolution of functions, &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;μ&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;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are positive constants that depends solely on &lt;em&gt;N&lt;/em&gt; and &lt;em&gt;μ&lt;/em&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;ξ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; that is positive and radially symmetric.&lt;/div&gt;&lt;div&gt;Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:&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;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 64-104"},"PeriodicalIF":2.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142702721","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":"Sturmian measures can be sublinearly approximated by periodic measures","authors":"Xiangtong Wang ,&nbsp;Liqi Zheng","doi":"10.1016/j.jde.2024.11.015","DOIUrl":"10.1016/j.jde.2024.11.015","url":null,"abstract":"<div><div>To study the approximation rate of an ergodic measure by periodic measures with respect to the Wasserstein distance, we introduce the concept of <em>τ</em>-uniquely ergodic measures, with <span><math><mi>τ</mi><mo>≥</mo><mn>0</mn></math></span>. We demonstrate that a <em>τ</em>-uniquely ergodic Borel probability measure on a subshift of finite type can be approximated by periodic measures at a rate of <span><math><mi>o</mi><mo>(</mo><msubsup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>τ</mi></mrow></msubsup><mo>⁡</mo><mi>N</mi><mo>)</mo></math></span>. In particular, we show that a Sturmian measure, which is <em>τ</em>-uniquely ergodic for any <span><math><mi>τ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, can be approximated by periodic measures with a sublinear rate.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 56-96"},"PeriodicalIF":2.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703308","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":"Structural stability of non-isentropic Euler-Poisson system for gaseous stars","authors":"Ben Duan ,&nbsp;Zhen Luo ,&nbsp;Chunpeng Wang","doi":"10.1016/j.jde.2024.11.010","DOIUrl":"10.1016/j.jde.2024.11.010","url":null,"abstract":"<div><div>This paper concerns the non-isentropic steady compressible Euler-Poisson system in annuluses, which models the motion of gaseous stars with the gravitational interactions between gas particles and pressure forces. In the paper, the Euler-Poisson system is reformulated and decomposed into transport equations and coupled second-order nonlinear elliptic equations in polar coordinates. Not only the existence and the uniqueness, but also the structural stability of subsonic solutions are established.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 105-131"},"PeriodicalIF":2.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704199","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 and continuity of random exponential attractors for stochastic 3D globally modified non-autonomous Navier-Stokes equation","authors":"Zongfei Han ,&nbsp;Shengfan Zhou","doi":"10.1016/j.jde.2024.11.014","DOIUrl":"10.1016/j.jde.2024.11.014","url":null,"abstract":"<div><div>In this paper, we deal with four problems. (i) Based on criteria for a continuous non-autonomous deterministic dynamical system (NDDS) and a continuous non-autonomous random dynamical system (NRDS), we construct an exponential attractor for a continuous NDDS and a family of random exponential attractors for a family of continuous non-autonomous random dynamical systems (NRDS), respectively. (ii) We prove that this family of random exponential attractors is continuous (or stable, robust, i.e., upper and lower semi-continuous) in the sense of the symmetric Hausdorff distance as the intensity of stochastic perturbations approaches zero. (iii) We prove that for two conjugate NRDS, if one has a random exponential attractor, then the other has a random exponential attractor, and that for two families of conjugate NRDS, if a family of random exponential attractors for one family is continuous, then a corresponding family of random exponential attractors for the other family is continuous. (iv) We apply our abstract result to study the existence and continuity of random exponential attractors for 3D globally modified non-autonomous Navier-Stokes equation with additive noise.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 1-55"},"PeriodicalIF":2.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703307","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":"Dynamics of vortex cap solutions on the rotating unit sphere","authors":"Claudia García ,&nbsp;Zineb Hassainia ,&nbsp;Emeric Roulley","doi":"10.1016/j.jde.2024.11.012","DOIUrl":"10.1016/j.jde.2024.11.012","url":null,"abstract":"<div><div>In this work, we analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured in <span><span>[28]</span></span>, <span><span>[29]</span></span>, <span><span>[60]</span></span>, <span><span>[61]</span></span>. Such solutions are piecewise constant vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis. The proof is based on the bifurcation from zonal solutions given by spherical caps. For the one–interface case, the bifurcation eigenvalues correspond to Burbea's frequencies obtained in the planar case but shifted by the rotation speed of the sphere. The two–interfaces case (also called band type or strip type solutions) is more delicate. Though, for any fixed large enough symmetry, and under some non-degeneracy conditions to avoid spectral collisions, we achieve the existence of at most two branches of bifurcation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"417 ","pages":"Pages 1-63"},"PeriodicalIF":2.4,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142702722","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":"Incompressible limit of all-time solutions to isentropic Navier-Stokes equations with ill-prepared data in bounded domains","authors":"Yaobin Ou ,&nbsp;Lu Yang","doi":"10.1016/j.jde.2024.11.009","DOIUrl":"10.1016/j.jde.2024.11.009","url":null,"abstract":"<div><div>In this paper, we study the incompressible limit of <em>all-time</em> strong solutions to the isentropic compressible Navier-Stokes equations with <em>ill-prepared</em> initial data and slip boundary condition in three-dimensional bounded domains. The uniform estimates with respect to both the Mach number <span><math><mi>ϵ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and all time <span><math><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span> are derived by establishing a nonlinear integral inequality. In contrast to previous results for well-prepared initial data, the time derivatives of the velocity are unbounded which leads to the loss of strong convergence of the velocity. The novelties of this paper are to establish weighted energy estimates of new-type and to carefully combine the estimates for the fast variables and the slow variables, especially for the highest-order spatial derivatives of the fast variables. The convergence to the global strong solution of incompressible Navier-Stokes equations is shown by applying the Helmoltz decomposition and the strong convergence of the incompressible part of the velocity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2293-2323"},"PeriodicalIF":2.4,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706885","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":"Long time behaviour of solutions to non-local and non-linear dispersal problems","authors":"Maciej Tadej","doi":"10.1016/j.jde.2024.10.046","DOIUrl":"10.1016/j.jde.2024.10.046","url":null,"abstract":"<div><div>This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the <em>n</em>-dimensional Euclidean space or a periodic environment modelled by <em>n</em>-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2043-2064"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652798","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":"Solving Riemann problems with a topological tool","authors":"Cesar S. Eschenazi ,&nbsp;Wanderson J. Lambert ,&nbsp;Marlon M. López-Flores ,&nbsp;Dan Marchesin ,&nbsp;Carlos F.B. Palmeira ,&nbsp;Bradley J. Plohr","doi":"10.1016/j.jde.2024.11.002","DOIUrl":"10.1016/j.jde.2024.11.002","url":null,"abstract":"<div><div>In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2134-2174"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652716","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":"Up to the first two order Melnikov analysis for the exact cyclicity of planar piecewise linear vector fields with nonlinear switching curve","authors":"Liqin Zhao,&nbsp;Zheng Si,&nbsp;Ranran Jia","doi":"10.1016/j.jde.2024.11.007","DOIUrl":"10.1016/j.jde.2024.11.007","url":null,"abstract":"<div><div>In this paper, we focus on providing the exact bounds for the maximum number of limit cycles <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> that planar piecewise linear differential systems with two zones separated by the curve <span><math><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> under perturbation of arbitrary polynomials of <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> with degree <em>n</em> can have, where <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. By the first two order Melnikov functions, we achieve that <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>12</mn></math></span>, <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for <span><math><mn>3</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>88</mn></math></span> and <span><math><mi>Z</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>n</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></span> for any <em>n</em>. The results are novel and improve the previous results in the literature.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 2255-2292"},"PeriodicalIF":2.4,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652717","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}