Pub Date : 2026-06-05Epub Date: 2026-02-11DOI: 10.1016/j.jde.2026.114202
Ben-Xing Zhou , Qinglong Zhou
In this paper, we study the relative Morse index theory of discrete nonlinear Schrödinger equations with strongly indefinite potential functions satisfying . As applications, we study the existence and multiplicity of homoclinic solutions for discrete asymptotically linear Schrödinger equations with saturable nonlinearity . In previous works, the prevalent assumption was confined to coercive potential functions (satisfying ), in contrast to the strongly indefinite potential functions considered herein (with ).
{"title":"Relative Morse index of the discrete nonlinear Schrödinger equations with strongly indefinite potentials and applications","authors":"Ben-Xing Zhou , Qinglong Zhou","doi":"10.1016/j.jde.2026.114202","DOIUrl":"10.1016/j.jde.2026.114202","url":null,"abstract":"<div><div>In this paper, we study the relative Morse index theory of discrete nonlinear Schrödinger equations<span><span><span><math><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>ω</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span></span></span> with strongly indefinite potential functions <span><math><mi>V</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></math></span> satisfying <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mo>|</mo><mi>n</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><mo>|</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><mo>=</mo><mo>+</mo><mo>∞</mo></math></span>. As applications, we study the existence and multiplicity of homoclinic solutions for discrete asymptotically linear Schrödinger equations with saturable nonlinearity <span><math><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></math></span>. In previous works, the prevalent assumption was confined to coercive potential functions (satisfying <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mo>|</mo><mi>n</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mo>+</mo><mo>∞</mo></math></span>), in contrast to the strongly indefinite potential functions considered herein (with <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mo>|</mo><mi>n</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><mo>|</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><mo>=</mo><mo>+</mo><mo>∞</mo></math></span>).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114202"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147464","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}
Pub Date : 2026-06-05Epub Date: 2026-02-11DOI: 10.1016/j.jde.2026.114197
Károly J. Böröczky , Ágnes Kovács , Stephanie Mui , Gaoyong Zhang
This paper studies the general dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general dual Minkowski problem of prescribing the dual curvature measure of convex bodies. It is a Monge-Ampère type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative p using a variational method. This work generalizes recent results on the dual Minkowski problem of origin-symmetric convex bodies.
{"title":"Dual curvature density equation with group symmetry","authors":"Károly J. Böröczky , Ágnes Kovács , Stephanie Mui , Gaoyong Zhang","doi":"10.1016/j.jde.2026.114197","DOIUrl":"10.1016/j.jde.2026.114197","url":null,"abstract":"<div><div>This paper studies the general <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> dual Minkowski problem of prescribing the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> dual curvature measure of convex bodies. It is a Monge-Ampère type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative <em>p</em> using a variational method. This work generalizes recent results on the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> dual Minkowski problem of origin-symmetric convex bodies.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114197"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191964","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}
Pub Date : 2026-06-05Epub Date: 2026-02-12DOI: 10.1016/j.jde.2026.114203
Ciprian G. Gal , Maoyin Lv , Hao Wu
We examine a thermodynamically consistent diffuse interface model for two-phase incompressible viscous flows in a smooth bounded domain (). This model characterizes the evolution of free interfaces in contact with the solid boundary, specifically addressing the phenomenon of moving contact lines. The associated evolution system comprises a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity v, nonlinearly coupled with a convective Cahn–Hilliard equation governing the order parameter φ. Notably, for the boundary dynamics, the current model incorporates surface diffusion, a variable contact angle between the diffuse interface and the solid boundary, as well as mass transfer between bulk and surface. This material transfer adheres to a mass conservation law encompassing both bulk and surface contributions. In the general scenario of non-matched densities, we establish the existence of global weak solutions with finite energy in both two and three dimensions.
{"title":"On a thermodynamically consistent diffuse interface model for two-phase incompressible flows with non-matched densities: Dynamics of moving contact lines, surface diffusion, and mass transfer","authors":"Ciprian G. Gal , Maoyin Lv , Hao Wu","doi":"10.1016/j.jde.2026.114203","DOIUrl":"10.1016/j.jde.2026.114203","url":null,"abstract":"<div><div>We examine a thermodynamically consistent diffuse interface model for two-phase incompressible viscous flows in a smooth bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> (<span><math><mi>d</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>). This model characterizes the evolution of free interfaces in contact with the solid boundary, specifically addressing the phenomenon of moving contact lines. The associated evolution system comprises a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity <strong>v</strong>, nonlinearly coupled with a convective Cahn–Hilliard equation governing the order parameter <em>φ</em>. Notably, for the boundary dynamics, the current model incorporates surface diffusion, a variable contact angle between the diffuse interface and the solid boundary, as well as mass transfer between bulk and surface. This material transfer adheres to a mass conservation law encompassing both bulk and surface contributions. In the general scenario of non-matched densities, we establish the existence of global weak solutions with finite energy in both two and three dimensions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114203"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192802","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}
Pub Date : 2026-06-05Epub Date: 2026-02-13DOI: 10.1016/j.jde.2026.114204
Kexin Li , Xiaojing Xu
We are concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system with degenerate heat-conductivity describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. Both the specific volume and temperature are proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.
{"title":"Large-time behavior of solutions to compressible Navier-Stokes system in unbounded domains with degenerate heat-conductivity and large data","authors":"Kexin Li , Xiaojing Xu","doi":"10.1016/j.jde.2026.114204","DOIUrl":"10.1016/j.jde.2026.114204","url":null,"abstract":"<div><div>We are concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system with degenerate heat-conductivity describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. Both the specific volume and temperature are proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114204"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192742","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}
Pub Date : 2026-06-05Epub Date: 2026-02-12DOI: 10.1016/j.jde.2026.114220
Dawid Czapla , Sander C. Hille , Katarzyna Horbacz , Hanna Wojewódka-Ściążko
In this paper, we establish the law of the iterated logarithm for a wide class of non-stationary, continuous-time Markov processes evolving on Polish spaces. Specifically, our result applies to certain additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein distance, provided that a suitable moment condition involving the initial distribution is satisfied. Furthermore, we outline the application of this result to a Markov process arising as the solution of an infinite-dimensional stochastic differential equation with dissipative drift and additive noise.
{"title":"Law of the iterated logarithm for Markov semigroups with exponential mixing in the Wasserstein distance","authors":"Dawid Czapla , Sander C. Hille , Katarzyna Horbacz , Hanna Wojewódka-Ściążko","doi":"10.1016/j.jde.2026.114220","DOIUrl":"10.1016/j.jde.2026.114220","url":null,"abstract":"<div><div>In this paper, we establish the law of the iterated logarithm for a wide class of non-stationary, continuous-time Markov processes evolving on Polish spaces. Specifically, our result applies to certain additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein distance, provided that a suitable moment condition involving the initial distribution is satisfied. Furthermore, we outline the application of this result to a Markov process arising as the solution of an infinite-dimensional stochastic differential equation with dissipative drift and additive noise.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114220"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192743","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}
Pub Date : 2026-06-05Epub Date: 2026-02-12DOI: 10.1016/j.jde.2026.114207
Huagui Duan , Zhiping Fan
Let Σ be a compact non-degenerate star-shaped hypersurface in , we show the existence of at least five prime closed characteristics on Σ in two weak index settings. More precisely, we obtain the multiplicity under one of the following assumptions: (a) and ; (b) and , where is any prime closed characteristic on Σ and is any good m-th iteration.
{"title":"Multiple closed characteristics on compact star-shaped hypersurfaces in R10","authors":"Huagui Duan , Zhiping Fan","doi":"10.1016/j.jde.2026.114207","DOIUrl":"10.1016/j.jde.2026.114207","url":null,"abstract":"<div><div>Let Σ be a compact non-degenerate star-shaped hypersurface in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>10</mn></mrow></msup></math></span>, we show the existence of at least five prime closed characteristics on Σ in two weak index settings. More precisely, we obtain the multiplicity under one of the following assumptions: (a) <span><math><mi>i</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>≠</mo><mo>−</mo><mn>1</mn></math></span> and <span><math><mover><mrow><mi>i</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>y</mi><mo>)</mo><mo>≥</mo><mn>1</mn></math></span>; (b) <span><math><mi>i</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>≠</mo><mn>0</mn></math></span> and <span><math><mo>|</mo><mover><mrow><mi>i</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, where <span><math><mo>(</mo><mi>τ</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is any prime closed characteristic on Σ and <span><math><mo>(</mo><mi>m</mi><mi>τ</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is any good <em>m</em>-th iteration.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114207"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192805","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}
Pub Date : 2026-06-05Epub Date: 2026-02-12DOI: 10.1016/j.jde.2026.114218
Wei Wang, Meiqi Li
We study the repulsive chemotaxis-consumption system: , in bounded and smooth domains (), with no-flux and constant positive Dirichlet boundary conditions prescribed for u and v, respectively. Here D and S are suitably smooth and generalize the prototypes and with . When Ω is a ball, Wang and Winkler (2023) [20] established the finite-time blow-up of solutions for and . However, their proof cannot cover the seemingly inevitable blow-up for and , nor can it handle the possible finite-time blow-up in the more challenging case that the self-diffusion is relatively strong with . Essentially relying on the analysis of a novel moment-like functional tailored to superlinear sensitivity, we prove in this paper that if with , then for all initial data with sufficiently large mass, the corresponding initial-boundary value problem admits a finite-time blow-up solution. As opposed to the consideration for singularity formation, the global boundedness of solutions is also ascertained for .
{"title":"Finite-time blow-up and global boundedness in a repulsive chemotaxis-consumption system with general density-dependent sensitivity","authors":"Wei Wang, Meiqi Li","doi":"10.1016/j.jde.2026.114218","DOIUrl":"10.1016/j.jde.2026.114218","url":null,"abstract":"<div><div>We study the repulsive chemotaxis-consumption system: <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>+</mo><mfrac><mrow><mi>u</mi><mi>S</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mrow><mi>v</mi></mrow></mfrac><mi>∇</mi><mi>v</mi><mo>)</mo></math></span>, <span><math><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi></math></span> in bounded and smooth domains <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>), with no-flux and constant positive Dirichlet boundary conditions prescribed for <em>u</em> and <em>v</em>, respectively. Here <em>D</em> and <em>S</em> are suitably smooth and generalize the prototypes <span><math><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> and <span><math><mi>S</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>β</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>R</mi></math></span>. When Ω is a ball, Wang and Winkler (2023) <span><span>[20]</span></span> established the finite-time blow-up of solutions for <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>β</mi><mo>=</mo><mn>1</mn></math></span>. However, their proof cannot cover the seemingly inevitable blow-up for <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span> and <span><math><mi>β</mi><mo>></mo><mn>1</mn></math></span>, nor can it handle the possible finite-time blow-up in the more challenging case that the self-diffusion is relatively strong with <span><math><mi>α</mi><mo>≤</mo><mn>0</mn></math></span>. Essentially relying on the analysis of a novel moment-like functional tailored to superlinear sensitivity, we prove in this paper that if <span><math><mi>β</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>∩</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> with <span><math><mi>α</mi><mo>∈</mo><mi>R</mi></math></span>, then for all initial data with sufficiently large mass, the corresponding initial-boundary value problem admits a finite-time blow-up solution. As opposed to the consideration for singularity formation, the global boundedness of solutions is also ascertained for <span><math><mi>β</mi><mo><</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>−</mo><mi>α</mi></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114218"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192803","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}
Pub Date : 2026-06-05Epub Date: 2026-02-11DOI: 10.1016/j.jde.2026.114201
Kyeongsu Choi, Jiuzhou Huang
We classify the smooth self-similar solutions of the semilinear heat equation in satisfying an integral condition for all with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with and converges to a positive constant after rescaling at the blow-up point for all .
{"title":"Self-similar solutions of semilinear heat equations with positive speed","authors":"Kyeongsu Choi, Jiuzhou Huang","doi":"10.1016/j.jde.2026.114201","DOIUrl":"10.1016/j.jde.2026.114201","url":null,"abstract":"<div><div>We classify the smooth self-similar solutions of the semilinear heat equation <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></math></span> satisfying an integral condition for all <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span> with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with <span><math><mi>u</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>≥</mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo><mo>≥</mo><mn>0</mn></math></span> converges to a positive constant after rescaling at the blow-up point for all <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114201"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147465","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}
In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast system after cylindrical blow-up and use a well-known connection between the divergence integral along orbits and transition maps for vector fields. Since properties of the divergence integral depend on the location and multiplicity of singularities, we divide the sliding cycles into different classes, which can then produce different types of cyclicity results. As an example, we apply our results to regularized piecewise linear systems.
{"title":"Cyclicity of sliding cycles with singularities of regularized piecewise smooth visible-invisible two-folds","authors":"Jicai Huang , Renato Huzak , Otavio Henrique Perez , Jinhui Yao","doi":"10.1016/j.jde.2026.114205","DOIUrl":"10.1016/j.jde.2026.114205","url":null,"abstract":"<div><div>In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast system after cylindrical blow-up and use a well-known connection between the divergence integral along orbits and transition maps for vector fields. Since properties of the divergence integral depend on the location and multiplicity of singularities, we divide the sliding cycles into different classes, which can then produce different types of cyclicity results. As an example, we apply our results to regularized piecewise linear systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114205"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192806","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}
Pub Date : 2026-06-05Epub Date: 2026-02-12DOI: 10.1016/j.jde.2026.114216
Gaozhan Li , Lei Liu , Yiling Yang
In this paper, we consider the Cauchy problem for the focusing nonlinear Schrödinger (NLS) equation with a step-like background. Based on a proposed Riemann-Hilbert (RH) representation of the Cauchy problem, with nonlinear steepest descent method and a double limit technique, we derive the long-time behavior to the solution of the focusing NLS equation in a transition space-time region with near 0. Especially we find that the sub-leading term can be described by the solution of a fourth-order Painlevé transcendent, which is a modified version of the second member of generalized Painlevé II hierarchy. The numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.
{"title":"Complex Painlevé type transient asymptotics of the focusing NLS equation: Step-like oscillating background","authors":"Gaozhan Li , Lei Liu , Yiling Yang","doi":"10.1016/j.jde.2026.114216","DOIUrl":"10.1016/j.jde.2026.114216","url":null,"abstract":"<div><div>In this paper, we consider the Cauchy problem for the focusing nonlinear Schrödinger (NLS) equation with a step-like background. Based on a proposed Riemann-Hilbert (RH) representation of the Cauchy problem, with nonlinear steepest descent method and a double limit technique, we derive the long-time behavior to the solution of the focusing NLS equation in a transition space-time region with <span><math><mi>x</mi><mo>/</mo><mi>t</mi></math></span> near 0. Especially we find that the sub-leading term can be described by the solution of a fourth-order Painlevé transcendent, which is a modified version of the second member of generalized Painlevé II hierarchy. The numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"465 ","pages":"Article 114216"},"PeriodicalIF":2.3,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192804","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}
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