Carlos García-Azpeitia , Ziad Ghanem , Wiesław Krawcewicz
{"title":"Global bifurcation in symmetric systems of nonlinear wave equations","authors":"Carlos García-Azpeitia , Ziad Ghanem , Wiesław Krawcewicz","doi":"10.1016/j.jde.2025.113600","DOIUrl":"10.1016/j.jde.2025.113600","url":null,"abstract":"<div><div>In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and non-trivial damping. As a motivating example, we consider an application of our result to a system of <em>N</em> identical vibrating strings with dihedral coupling relations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113600"},"PeriodicalIF":2.4,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144589049","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":"Sample path properties and small ball probabilities for stochastic fractional diffusion equations","authors":"Yuhui Guo , Jian Song , Ran Wang , Yimin Xiao","doi":"10.1016/j.jde.2025.113604","DOIUrl":"10.1016/j.jde.2025.113604","url":null,"abstract":"<div><div>We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:<span><span><span><math><msup><mrow><mo>∂</mo></mrow><mrow><mi>β</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mn>0</mn><mo>+</mo></mrow><mrow><mi>γ</mi></mrow></msubsup><mrow><mo>[</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><mi>γ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> is the fractional/power of Laplacian and <span><math><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is a fractional space-time Gaussian noise. We prove the existence and uniqueness of the solution and then focus on various sample path regularity properties of the solution. More specifically, we establish the exact uniform and local moduli of continuity and Chung's laws of the iterated logarithm. The small ball probability is also studied.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113604"},"PeriodicalIF":2.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579418","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":"Inverse scattering problem for operators with a finite-dimensional non-local potential","authors":"V.A. Zolotarev","doi":"10.1016/j.jde.2025.113606","DOIUrl":"10.1016/j.jde.2025.113606","url":null,"abstract":"<div><div>Scattering problem for a self-adjoint integro-differential operator, which is the sum of the operator of the second derivative and of a finite-dimensional self-adjoint operator, is studied. Jost solutions are found and it is shown that the scattering function has a multiplicative structure, besides, each of the multipliers is a scattering coefficient for a pair of self-adjoint operators, one of which is a one-dimensional perturbation of the other. Solution of the inverse problem is based upon the solutions to the inverse problem for every multiplier. A technique for finding parameters of the finite-dimensional perturbation via the scattering data is described.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113606"},"PeriodicalIF":2.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579420","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}
Ryan Alvarado , Przemysław Górka , Artur Słabuszewski
{"title":"Compact embeddings of Sobolev, Besov, and Triebel–Lizorkin spaces","authors":"Ryan Alvarado , Przemysław Górka , Artur Słabuszewski","doi":"10.1016/j.jde.2025.113598","DOIUrl":"10.1016/j.jde.2025.113598","url":null,"abstract":"<div><div>We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel–Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113598"},"PeriodicalIF":2.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579419","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":"Uniqueness & weak-BV stability in the large for isothermal gas dynamics","authors":"Jeffrey Cheng","doi":"10.1016/j.jde.2025.113599","DOIUrl":"10.1016/j.jde.2025.113599","url":null,"abstract":"<div><div>For the 1-d isothermal Euler system, we consider the family of entropic BV solutions with possibly large, but finite, total variation. We show that these solutions are stable with respect to large perturbations in a class of weak solutions to the system which may not even be BV. The method is based on the construction of a modified front tracking algorithm, in which the theory of <em>a</em>-contraction with shifts for shocks is used as a building block. The main contribution is to construct the weight in the modified front tracking algorithm in a large-BV setting.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113599"},"PeriodicalIF":2.4,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571354","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":"Principal eigenvalue theory of nonlocal dispersal cooperative systems in unbounded domains and applications","authors":"Hao Wu, Yan-Xia Feng, Wan-Tong Li, Jian-Wen Sun","doi":"10.1016/j.jde.2025.113592","DOIUrl":"10.1016/j.jde.2025.113592","url":null,"abstract":"<div><div>This paper is concerned with the theory of principal eigenvalue of a class of nonlocal dispersal cooperative systems in unbounded domains. We begin by establishing a new Harnack inequality for positive solutions of the nonlocal dispersal system. Using this inequality, we derive a sufficient condition for the existence of principal eigenvalues in unbounded domains. Additionally, we construct a matrix-valued sequence that approximates the nonlocal dispersal system and meets the sufficient condition. We then investigate the relationships among various definitions of generalized principal eigenvalues in both bounded and unbounded domains. A detailed analysis is conducted on how the principal eigenvalue impacts the effectiveness of the maximum principle. Finally, we apply our principal eigenvalue theory to examine the nonlocal dispersal cooperative system in unbounded domains and obtain the global dynamics of two vector-host epidemic models.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113592"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571187","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":"Wavefronts for a degenerate reaction-diffusion system with application to bacterial growth models","authors":"Luisa Malaguti, Elisa Sovrano","doi":"10.1016/j.jde.2025.113593","DOIUrl":"10.1016/j.jde.2025.113593","url":null,"abstract":"<div><div>We investigate wavefront solutions in a nonlinear system of two coupled reaction-diffusion equations with degenerate diffusivity:<span><span><span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>−</mo><mi>n</mi><mi>b</mi><mo>,</mo><mspace></mspace><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mo>[</mo><mi>D</mi><mi>n</mi><mi>b</mi><msub><mrow><mi>b</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>]</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi>n</mi><mi>b</mi><mo>,</mo></math></span></span></span> where <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span>, and <em>D</em> is a positive diffusion coefficient. This model, introduced by Kawasaki et al. (1997) <span><span>[3]</span></span>, describes the spatial-temporal dynamics of bacterial colonies <span><math><mi>b</mi><mo>=</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> and nutrients <span><math><mi>n</mi><mo>=</mo><mi>n</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> on agar plates. While Kawasaki et al. provided numerical evidence for wavefronts, analytical confirmation remained an open problem. We prove the existence of an infinite family of wavefronts parameterized by their wave speed, which varies on a closed positive half-line. We provide an upper bound for the threshold speed and a lower bound for it when <em>D</em> is sufficiently large. The proofs are based on several analytical tools, including the shooting method and the fixed-point theory in Fréchet spaces, to establish existence, and the central manifold theorem to ascertain uniqueness.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113593"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570047","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}
Marta García-Huidobro , Raúl Manásevich , Jean Mawhin , Satoshi Tanaka
{"title":"Corrigendum to “Periodic solutions for nonlinear systems of Ode's with generalized variable exponents operators” [J. Differ. Equ. 388 (2024) 34–58]","authors":"Marta García-Huidobro , Raúl Manásevich , Jean Mawhin , Satoshi Tanaka","doi":"10.1016/j.jde.2025.113601","DOIUrl":"10.1016/j.jde.2025.113601","url":null,"abstract":"<div><div>We provide two ways to overcome a minor gap in the proof of Lemma 6.1 of <span><span>[1]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113601"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632435","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 existence for multi-dimensional partially diffusive systems","authors":"Jean-Paul Adogbo, Raphäel Danchin","doi":"10.1016/j.jde.2025.113596","DOIUrl":"10.1016/j.jde.2025.113596","url":null,"abstract":"<div><div>In this work, we explore the global existence of strong solutions for a class of partially diffusive hyperbolic systems within the framework of critical homogeneous Besov spaces. Our objective is twofold: first, to extend our recent findings on the local existence presented in <span><span>[1]</span></span>, and second, to refine and enhance the analysis of Kawashima <span><span>[15]</span></span>.</div><div>To address the distinct behaviors of low and high frequency regimes, we employ a hybrid Besov norm approach that incorporates different regularity exponents for each regime. This allows us to meticulously analyze the interactions between these regimes, which exhibit fundamentally different dynamics.</div><div>A significant part of our methodology is based on the study of a Lyapunov functional, inspired by the work of Beauchard and Zuazua <span><span>[3]</span></span> and recent contributions <span><span>[8]</span></span>, <span><span>[7]</span></span>, <span><span>[6]</span></span>. To effectively handle the high-frequency components, we introduce a parabolic mode with better smoothing properties, which plays a central role in our analysis.</div><div>Our results are particularly relevant for important physical systems, such as the magnetohydrodynamics (MHD) system and the barotropic compressible Navier-Stokes equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"445 ","pages":"Article 113596"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571186","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}
Laura Baldelli , Jarosław Mederski , Alessio Pomponio
{"title":"Normalized solutions to quasilinear problems involving Born-Infeld-type operators","authors":"Laura Baldelli , Jarosław Mederski , Alessio Pomponio","doi":"10.1016/j.jde.2025.113595","DOIUrl":"10.1016/j.jde.2025.113595","url":null,"abstract":"<div><div>The paper concerns the existence of normalized solutions to a large class of quasilinear problems, including the well-known Born-Infeld operator. In the mass subcritical cases, we study a global minimization problem and obtain a ground state solution for a <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-type operator which implies the existence of solutions to the Born-Infeld problem. We also deal with the mass critical and mass supercritical cases for quasilinear problems involving the <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-type operator.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113595"},"PeriodicalIF":2.4,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570048","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}