Journal of Differential Equations最新文献

{"title":"Multiple solutions for quasi-linear elliptic equations with lack of symmetry","authors":"Chen Huang ,&nbsp;Youjun Wang ,&nbsp;Jianjun Zhang","doi":"10.1016/j.jde.2026.114177","DOIUrl":"10.1016/j.jde.2026.114177","url":null,"abstract":"<div><div>This paper investigates the existence of multiple solutions for a class of generalized quasi-linear elliptic equations on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with lack of symmetry. The primary challenges in addressing such problem stem from the loss of smoothness and compactness in the associated energy functional. To overcome these obstacles, we introduce a variational perturbation method inspired by the approach developed by Liu-Liu-Wang (2019) <span><span>[23]</span></span>. Subsequently, by employing an abstract critical point theorem along with Moser's iteration technique, we establish the existence of arbitrarily many critical points for the corresponding non-smooth and non-even functionals.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114177"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146170549","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":"Variational construction of asymptotic orbits in contact Hamiltonian systems","authors":"Liang Jin ,&nbsp;Jun Yan ,&nbsp;Kai Zhao","doi":"10.1016/j.jde.2026.114141","DOIUrl":"10.1016/j.jde.2026.114141","url":null,"abstract":"<div><div>For contact Hamiltonian systems without monotonicity assumption, there is a family of invariant sets <span><math><mo>{</mo><msub><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>u</mi></mrow></msub><mo>}</mo></math></span> naturally stratified by the solutions <em>u</em> to the corresponding Hamilton-Jacobi equation. Under convergence assumptions of the solution semigroup, we establish the existence of semi-infinite orbits asymptotic to some <span><math><msub><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>u</mi></mrow></msub></math></span> and heteroclinic orbits between <span><math><msub><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>u</mi></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>v</mi></mrow></msub></math></span> for different solutions <em>u</em> and <em>v</em> by variational methods. We also give verifiable criteria to ensure the convergence assumptions. As a corollary, we give a description of action minimizing orbits of the model system studied in <span><span>[26]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114141"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025659","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":"High multiplicity and global structure of coexistence states in a predator-prey model with saturation","authors":"Kousuke Kuto ,&nbsp;Julián López-Gómez ,&nbsp;Eduardo Muñoz-Hernández","doi":"10.1016/j.jde.2026.114116","DOIUrl":"10.1016/j.jde.2026.114116","url":null,"abstract":"<div><div>This paper establishes that, under the appropriate range of values of the parameters involved in the formulation of the model, a diffusive predator-prey system with saturation can have an arbitrarily large number of coexistence states for sufficiently large saturation rates. Moreover, it ascertains the global structure of the set of coexistence states in the limiting system as the saturation rate blows up.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114116"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001827","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 strongly nonlocal Allen–Cahn problem","authors":"Erisa Hasani,&nbsp;Stefania Patrizi","doi":"10.1016/j.jde.2026.114168","DOIUrl":"10.1016/j.jde.2026.114168","url":null,"abstract":"<div><div>We study the sharp interface limit of the fractional Allen–Cahn equation<span><span><span><math><mi>ε</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>=</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>[</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>]</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msup></mrow></mfrac><msup><mrow><mi>W</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>)</mo><mspace></mspace><mtext>in</mtext><mspace></mspace><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo></math></span></span></span> where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span>, <span><math><msubsup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>=</mo><mo>−</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>s</mi></mrow></msub><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> is the fractional Laplacian of order <span><math><mn>2</mn><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and <em>W</em> is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution <span><math><msup><mrow><mi>u</mi></mrow><mrow><mi>ε</mi></mrow></msup></math></span> converges, as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>, to the minima of <em>W</em> with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for <span><math><mi>s</mi><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114168"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075449","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 of variational solutions to doubly nonlinear systems in general noncylindrical domains","authors":"Leah Schätzler ,&nbsp;Christoph Scheven ,&nbsp;Jarkko Siltakoski ,&nbsp;Calvin Stanko","doi":"10.1016/j.jde.2026.114139","DOIUrl":"10.1016/j.jde.2026.114139","url":null,"abstract":"<div><div>We consider the Cauchy–Dirichlet problem to doubly nonlinear systems of the form<span><span><span><math><mrow><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>−</mo><mi>div</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>ξ</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo><mo>)</mo><mo>=</mo><mo>−</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>u</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>)</mo></mrow></math></span></span></span> with <span><math><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> in a bounded noncylindrical domain <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Further, we suppose that <span><math><mi>x</mi><mo>↦</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></math></span> is integrable, that <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo><mo>↦</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></math></span> is convex, and that <em>f</em> satisfies a <em>p</em>-growth and -coercivity condition for some <span><math><mi>p</mi><mo>&gt;</mo><mi>max</mi><mo>⁡</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>}</mo></mrow></math></span>. Merely assuming that <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>E</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, we prove the existence of variational solutions <span><math><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>E</mi><mo>,</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span>. If <em>E</em> does not shrink too fast, we show that for the solution <em>u</em> constructed in the first step, <span><math><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi></math></span> admits a distributional time derivative. Moreover, under suitable conditions on <em>E</em> and the stricter lower bound <span><math><mi>p</mi><mo>≥</mo><mfrac><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span>, <em>u</em> is continuous with respect to time.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114139"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075451","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":"Acceleration or finite speed propagation in integro-differential equations with logarithmic Allee effects","authors":"Émeric Bouin ,&nbsp;Jérôme Coville ,&nbsp;Xi Zhang","doi":"10.1016/j.jde.2026.114136","DOIUrl":"10.1016/j.jde.2026.114136","url":null,"abstract":"<div><div>This paper is devoted to studying propagation phenomena in integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity and the standard weak Allee effect one. We study the effect of the tails of the dispersal kernel on the rate of expansion. When the tail of the kernel is sub-exponential, the exact separation between existence and non-existence of travelling waves is exhibited. This, in turn, provides the exact separation between finite speed propagation and acceleration in the Cauchy problem. Moreover, the exact rates of acceleration for dispersal kernels with sub-exponential and algebraic tails are provided. Our approach is generic and covers a large variety of dispersal kernels including those leading to convolution and fractional Laplace operators. Numerical simulations are provided to illustrate our results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114136"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075455","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 blow-up via renormalized Kato theory: Resolving Nakao-type systems","authors":"Mengyun Liu","doi":"10.1016/j.jde.2026.114165","DOIUrl":"10.1016/j.jde.2026.114165","url":null,"abstract":"<div><div>We address the fundamental obstruction identified in <span><span>[10, Remark 3]</span></span> for system (1) where sign-changing kernels when <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&lt;</mo><mn>4</mn><mi>k</mi></math></span> preclude blow-up arguments via nonnegative functionals—by partially resolving it in the regime <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&lt;</mo><mi>k</mi></math></span>.</div><div>Building upon Kato-type techniques, we develop a renormalized iteration scheme establishing the quantitative upper bounds for blow-up times. This framework resolves the critical case <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&lt;</mo><mi>k</mi></math></span> for <span><math><mi>b</mi><mo>,</mo><mi>k</mi><mo>&gt;</mo><mn>0</mn></math></span> under <span><math><mi>θ</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≥</mo><mn>0</mn></math></span>. When combined with <span><span>[10]</span></span>'s results for <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≥</mo><mn>4</mn><mi>k</mi></math></span>, it completes the blow-up theory for the subregime <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>&lt;</mo><mi>k</mi></math></span>. For <span><math><mi>b</mi><mo>,</mo><mi>k</mi><mo>&lt;</mo><mn>0</mn></math></span>, we prove blow-up in the extended critical region<span><span><span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>G</mi><mi>G</mi></mrow></msub></mrow></msub><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mi>max</mi><mo>⁡</mo><mrow><mo>{</mo><mfrac><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>p</mi><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mspace></mspace><mfrac><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>p</mi><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>}</mo></mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≥</mo><mn>0</mn><mo>,</mo></math></span></span></span> strictly containing the classical critical set.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114165"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075456","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 well-posedness of (nonlinear) rough continuity equations","authors":"Lucio Galeati ,&nbsp;James-Michael Leahy ,&nbsp;Torstein Nilssen","doi":"10.1016/j.jde.2026.114124","DOIUrl":"10.1016/j.jde.2026.114124","url":null,"abstract":"<div><div>Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood drifts, as well as well-posedness of weak <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-valued solutions to linear rough continuity and transport equations on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> under DiPerna–Lions regularity conditions; a combination of the two then yields flow representation formulas for linear RPDEs. We apply these results to obtain existence, uniqueness and continuous dependence for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the 2D Euler equations in vorticity form with rough transport noise, providing a rough analogue of Yudovich's theorem. As a consequence, we construct an associated continuous random dynamical system, when the driving noise is a fractional Brownian motion with Hurst parameter <span><math><mi>H</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We further prove weak existence of solutions for initial vorticities in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>, for any <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114124"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146001826","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 asymptotic behavior of one-dimensional solutions in cylinders","authors":"Francesca De Marchis,&nbsp;Lisa Mazzuoli,&nbsp;Filomena Pacella","doi":"10.1016/j.jde.2026.114146","DOIUrl":"10.1016/j.jde.2026.114146","url":null,"abstract":"<div><div>We consider positive one-dimensional solutions of a Lane-Emden relative Dirichlet problem in a cylinder and study their stability/instability properties as the energy varies with respect to domain perturbations. This depends on the exponent <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span> of the nonlinearity and we obtain results for <em>p</em> close to 1 and for <em>p</em> large. This is achieved by a careful asymptotic analysis of the one-dimensional solution as <span><math><mi>p</mi><mo>→</mo><mn>1</mn></math></span> or <span><math><mi>p</mi><mo>→</mo><mo>∞</mo></math></span>, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114146"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075452","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 an on-off self-propelled particle in a cellular flow","authors":"Alfredo J. Grados,&nbsp;Jeferson Cassiano,&nbsp;Maurício F.S. Lima,&nbsp;Rafael D. Vilela","doi":"10.1016/j.jde.2026.114127","DOIUrl":"10.1016/j.jde.2026.114127","url":null,"abstract":"<div><div>On-off swimming mechanisms are common in natural and artificial microswimmers. A mathematically sound modeling of the dynamics of particles exhibiting such mechanisms should be based on discontinuous vector fields. Nevertheless, the relatively new techniques constructed to deal with discontinuous systems have not been employed for microswimmers. Here, motivated by the “run” strategy proposed in T. Mano et al. (2017) <span><span>[11]</span></span> for motile Janus particles, we study the dynamics of swimmers with self-propulsion when their orientation is close to a given direction. Unlike the dynamics addressed in the aforementioned reference, we consider that swimmers are immersed in a fluid flow. Their orientation, therefore, evolves according to Jeffery's equation. We use the Filippov formalism for discontinuous systems to geometrically describe the velocity field. We also derive a Poincaré map, which describes the dynamics to first order in the discontinuity parameter, and study some of its properties.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"462 ","pages":"Article 114127"},"PeriodicalIF":2.3,"publicationDate":"2026-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146075454","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}