Pub Date : 2026-04-05Epub Date: 2025-12-22DOI: 10.1016/j.jde.2025.114038
Changchang Guan , Shi-Liang Wu , Shigui Ruan
This paper is concerned with a time periodic Lotka-Volterra diffusion system with strong competition. We study the long time behavior of bounded solutions for the system that lie between two stable semi-trivial periodic solutions of the corresponding kinetic system. By transforming the competitive system into an equivalent cooperative system on , we first demonstrate local stability of a pair of diverging periodic traveling fronts. Then, by establishing a new Liouville-type theorem for solutions of the wave profile system and applying the truncation method, we prove asymptotic stability of these diverging periodic traveling fronts in the -norm. Based on this result, by investigating the behavior of solutions with a one-parameter family of initial data, we present the trichotomy of parameter-dependent solutions: propagation for large parameter values, extinction for small parameter values, and transition from propagation to extinction for intermediate parameter values. Finally, we explore some properties of the threshold solution.
{"title":"Long time behavior for a periodic Lotka-Volterra reaction-diffusion system with strong competition II: The threshold phenomenon","authors":"Changchang Guan , Shi-Liang Wu , Shigui Ruan","doi":"10.1016/j.jde.2025.114038","DOIUrl":"10.1016/j.jde.2025.114038","url":null,"abstract":"<div><div>This paper is concerned with a time periodic Lotka-Volterra diffusion system with strong competition. We study the long time behavior of bounded solutions for the system that lie between two stable semi-trivial periodic solutions of the corresponding kinetic system. By transforming the competitive system into an equivalent cooperative system on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, we first demonstrate local stability of a pair of diverging periodic traveling fronts. Then, by establishing a new Liouville-type theorem for solutions of the wave profile system and applying the truncation method, we prove asymptotic stability of these diverging periodic traveling fronts in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-norm. Based on this result, by investigating the behavior of solutions with a one-parameter family of initial data, we present the trichotomy of parameter-dependent solutions: propagation for large parameter values, extinction for small parameter values, and transition from propagation to extinction for intermediate parameter values. Finally, we explore some properties of the threshold solution.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114038"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838859","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 : 2026-04-05Epub Date: 2026-01-28DOI: 10.1016/j.jde.2026.114158
Jingyu Li , Xiaowen Li , Ming Mei
This paper is concerned with the large time behaviors of solutions to the Burgers equation of porous-media type in the form of , where the diffusion with possesses the strong singularity of fast-diffusion at . The main issue of the paper is to show the asymptotic stability of viscous shock profiles with the constant states , where the strong singularity exhibits for the equation when the viscous shock wave reaches the singular point . To overcome such a strong singularity for wave stability, we first need to analyze the rate of the viscous shock wave to , then we artfully choose some weight functions which are closely dependent on the decay rate of the viscous shock wave to the singular point , and further show the wave stability by the weighted-energy-method.
{"title":"Asymptotic stability of viscous shock profiles to Burgers equation with singular super-fast diffusion","authors":"Jingyu Li , Xiaowen Li , Ming Mei","doi":"10.1016/j.jde.2026.114158","DOIUrl":"10.1016/j.jde.2026.114158","url":null,"abstract":"<div><div>This paper is concerned with the large time behaviors of solutions to the Burgers equation of porous-media type in the form of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>f</mi><msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span>, where the diffusion <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><msub><mrow><mo>(</mo><mfrac><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mn>1</mn><mo>+</mo><mo>|</mo><mi>m</mi><mo>|</mo></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub></math></span> with <span><math><mi>m</mi><mo><</mo><mn>0</mn></math></span> possesses the strong singularity of fast-diffusion at <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>. The main issue of the paper is to show the asymptotic stability of viscous shock profiles with the constant states <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>></mo><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, where the strong singularity exhibits for the equation when the viscous shock wave reaches the singular point <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>. To overcome such a strong singularity for wave stability, we first need to analyze the rate of the viscous shock wave to <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, then we artfully choose some weight functions which are closely dependent on the decay rate of the viscous shock wave to the singular point <span><math><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mn>0</mn></math></span>, and further show the wave stability by the weighted-energy-method.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114158"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074031","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 : 2026-04-05Epub Date: 2026-01-14DOI: 10.1016/j.jde.2025.114091
Jing Gao , Weijun Zhang
<div><div>In this paper, we investigate bifurcation phenomena for fully nonlinear elliptic equations and coupled systems dominated by <em>k</em>-Hessian operator. Specifically, we consider the Dirichlet problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span></div><div>as well as the coupled system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>g</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>v</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>h</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> where Ω is a bounded strictly (<em>k</em>-1)-convex domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>λ</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> is a continuous function with zeros only at 0 and <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> are continuous functions with zeros only at <span><math><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><mo
{"title":"Bifurcation on fully nonlinear elliptic equations and systems","authors":"Jing Gao , Weijun Zhang","doi":"10.1016/j.jde.2025.114091","DOIUrl":"10.1016/j.jde.2025.114091","url":null,"abstract":"<div><div>In this paper, we investigate bifurcation phenomena for fully nonlinear elliptic equations and coupled systems dominated by <em>k</em>-Hessian operator. Specifically, we consider the Dirichlet problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span></div><div>as well as the coupled system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>g</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>v</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></msup><mo>=</mo><mi>λ</mi><mi>h</mi><mo>(</mo><mo>−</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo><mspace></mspace></mtd><mtd><mi>i</mi><mi>n</mi><mspace></mspace><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>v</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mi>o</mi><mi>n</mi><mspace></mspace><mo>∂</mo><mi>Ω</mi></mtd></mtr></mtable></mrow></math></span></span></span> where Ω is a bounded strictly (<em>k</em>-1)-convex domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>λ</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> is a continuous function with zeros only at 0 and <span><math><mi>g</mi><mo>,</mo><mi>h</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow><mo>→</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></mrow></math></span> are continuous functions with zeros only at <span><math><mo>(</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><mo","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114091"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978284","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 : 2026-04-05Epub Date: 2025-12-22DOI: 10.1016/j.jde.2025.114051
Rajendra Beekie , Siming He
We study the stability of spectrally stable, strictly monotone, and smooth shear flows in the 2D Navier-Stokes equations on with small viscosity ν. We establish nonlinear stability in for with a threshold of size for time smaller than with . Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.
{"title":"Transition threshold for strictly monotone shear flows in Sobolev spaces","authors":"Rajendra Beekie , Siming He","doi":"10.1016/j.jde.2025.114051","DOIUrl":"10.1016/j.jde.2025.114051","url":null,"abstract":"<div><div>We study the stability of spectrally stable, strictly monotone, and smooth shear flows in the 2D Navier-Stokes equations on <span><math><mi>T</mi><mo>×</mo><mi>R</mi></math></span> with small viscosity <em>ν</em>. We establish nonlinear stability in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> for <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span> with a threshold of size <span><math><mi>ϵ</mi><msup><mrow><mi>ν</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> for time smaller than <span><math><msub><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msup><mrow><mi>ν</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> with <span><math><mi>ϵ</mi><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>≪</mo><mn>1</mn></math></span>. Additionally, we demonstrate nonlinear inviscid damping and enhanced dissipation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114051"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838861","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 : 2026-04-05Epub Date: 2026-01-08DOI: 10.1016/j.jde.2025.114083
Nguyen Huy Tuan , Nguyen Anh Tuan
This study analyzes a nonlocal-in-time Keller-Segel (KS) chemotaxis system describing organism movement with memory effects. Two distinct regimes are tackled. Firstly, for the time-fractional KS equation augmented by a logistic source, we show that sufficiently dominant damping guarantees existence of a unique global mild solution that remains uniformly bounded for all time. The proof blends a priori estimates in uniformly local Lebesgue spaces with new semigroup bounds for solution operators involving Mittag-Leffler kernels. Secondly, removing the logistic term, we investigate singular behavior. Via Fourier analysis and Besov-Triebel-Lizorkin embeddings we construct initial data leading to finite-time blowup. Additionally, Littlewood-Paley decompositions reveal norm inflation: arbitrarily small data in rough topologies can produce nonzero solution norms instantaneously, signaling ill-posedness. Together, these results shed light on open issues regarding the global boundedness and singular solutions for memory-driven chemotaxis system.
{"title":"Global dynamics of the nonlocal Keller-Segel system: Uniform boundedness and singular behavior","authors":"Nguyen Huy Tuan , Nguyen Anh Tuan","doi":"10.1016/j.jde.2025.114083","DOIUrl":"10.1016/j.jde.2025.114083","url":null,"abstract":"<div><div>This study analyzes a nonlocal-in-time Keller-Segel (KS) chemotaxis system describing organism movement with memory effects. Two distinct regimes are tackled. Firstly, for the time-fractional KS equation augmented by a logistic source, we show that sufficiently dominant damping guarantees existence of a unique global mild solution that remains uniformly bounded for all time. The proof blends a priori estimates in uniformly local Lebesgue spaces with new semigroup bounds for solution operators involving Mittag-Leffler kernels. Secondly, removing the logistic term, we investigate singular behavior. Via Fourier analysis and Besov-Triebel-Lizorkin embeddings we construct initial data leading to finite-time blowup. Additionally, Littlewood-Paley decompositions reveal norm inflation: arbitrarily small data in rough topologies can produce nonzero solution norms instantaneously, signaling ill-posedness. Together, these results shed light on open issues regarding the global boundedness and singular solutions for memory-driven chemotaxis system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114083"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145921895","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 : 2026-04-05Epub Date: 2026-01-28DOI: 10.1016/j.jde.2026.114160
Kaibin Zhang , Xinhua Li , Chunyou Sun
In this paper, we focus on the stability and long-time behavior problem for the 2D Boussinesq equations near the hydrostatic equilibrium with partial dissipation in the velocity and horizontal thermal diffusion. The lack of dissipation in the first component of the velocity and vertical thermal diffusion leads to the main difficulties. We establish the stability in , and demonstrate the exponential decay of its oscillatory portion in the .
{"title":"Stability and exponential decay for the 2D anisotropic Boussinesq equations near the hydrostatic equilibrium","authors":"Kaibin Zhang , Xinhua Li , Chunyou Sun","doi":"10.1016/j.jde.2026.114160","DOIUrl":"10.1016/j.jde.2026.114160","url":null,"abstract":"<div><div>In this paper, we focus on the stability and long-time behavior problem for the 2D Boussinesq equations near the hydrostatic equilibrium with partial dissipation in the velocity and horizontal thermal diffusion. The lack of dissipation in the first component of the velocity and vertical thermal diffusion leads to the main difficulties. We establish the stability in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and demonstrate the exponential decay of its oscillatory portion in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114160"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146074033","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 : 2026-04-05Epub Date: 2025-12-18DOI: 10.1016/j.jde.2025.114053
David Cheban
The aim of this paper is to study the problem of existence of asymptotically periodic solutions of the scalar differential equation , where is a continuous asymptotically τ-periodic function. We prove that every bounded on semi-axis solution φ of this equation is S-asymptotically τ-periodic, i.e., . This statement is a generalization of the well-known Massera's theorem for asymptotically periodic scalar differential equations. We also establish a similar statement for scalar difference equations.
{"title":"Massera's theorem for asymptotically periodic scalar differential equations","authors":"David Cheban","doi":"10.1016/j.jde.2025.114053","DOIUrl":"10.1016/j.jde.2025.114053","url":null,"abstract":"<div><div>The aim of this paper is to study the problem of existence of asymptotically periodic solutions of the scalar differential equation <span><math><msup><mrow><mi>x</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>, where <span><math><mi>f</mi><mo>:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> is a continuous asymptotically <em>τ</em>-periodic function. We prove that every bounded on semi-axis <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> solution <em>φ</em> of this equation is S-asymptotically <em>τ</em>-periodic, i.e., <span><math><munder><mi>lim</mi><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></munder><mo></mo><mo>|</mo><mi>φ</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>τ</mi><mo>)</mo><mo>−</mo><mi>φ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>0</mn></math></span>. This statement is a generalization of the well-known Massera's theorem for asymptotically periodic scalar differential equations. We also establish a similar statement for scalar difference equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114053"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771864","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 : 2026-04-05Epub Date: 2025-12-30DOI: 10.1016/j.jde.2025.114074
Qing Guo , Angela Pistoia , Shixin Wen
This paper deals with the existence of positive solutions to the system where , , , and is sufficiently small. The interaction coefficient as .
We construct a family of segregated solutions to this system, where each component blows-up at a different critical point of the Robin function as . The system lacks a variational formulation due to its specific coupling form, which leads to essentially different behaviors in the subcritical, critical, and supercritical regimes and requires an appropriate functional settings to carry out the construction.
{"title":"Segregated solutions for a class of systems with Lotka-Volterra interaction","authors":"Qing Guo , Angela Pistoia , Shixin Wen","doi":"10.1016/j.jde.2025.114074","DOIUrl":"10.1016/j.jde.2025.114074","url":null,"abstract":"<div><div>This paper deals with the existence of positive solutions to the system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mi>ε</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mi>β</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>ε</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></msubsup><mo>+</mo><mi>β</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>on </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <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>4</mn></math></span>, <span><math><mi>p</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span>, and <span><math><mi>ε</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>)</mo></math></span> is sufficiently small. The interaction coefficient <span><math><mi>β</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>ε</mi><mo>)</mo><mo>→</mo><mn>0</mn></math></span> as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>.</div><div>We construct a family of <em>segregated solutions</em> to this system, where each component blows-up at a different critical point of the Robin function as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>. The system lacks a variational formulation due to its specific coupling form, which leads to essentially different behaviors in the subcritical, critical, and supercritical regimes and requires an appropriate functional settings to carry out the construction.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114074"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145921894","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 : 2026-04-05Epub Date: 2025-12-22DOI: 10.1016/j.jde.2025.114036
Nicolas Augier , Milan Korda , Rodolfo Ríos-Zertuche
In this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state-constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. This example therefore demonstrates the impossibility of Filippov–Wažewski type approximation in the presence of state constraints. Our example also presents an occupation measure relaxation gap.
{"title":"Young measure relaxation gaps for controllable systems with smooth state constraints","authors":"Nicolas Augier , Milan Korda , Rodolfo Ríos-Zertuche","doi":"10.1016/j.jde.2025.114036","DOIUrl":"10.1016/j.jde.2025.114036","url":null,"abstract":"<div><div>In this article, we tackle the problem of the existence of a gap corresponding to Young measure relaxations for state-constrained optimal control problems. We provide a counterexample proving that a gap may occur in a very regular setting, namely for a smooth controllable system state-constrained to the closed unit ball, provided that the Lagrangian density (i.e., the running cost) is non-convex in the control variables. The example is constructed in the setting of sub-Riemannian geometry with the core ingredient being an unusual admissible curve that exhibits a certain form of resistance to state-constrained approximation. Specifically, this curve cannot be approximated by neighboring admissible curves while obeying the state constraint due to the intricate nature of the dynamics near the boundary of the constraint set. This example therefore demonstrates the impossibility of Filippov–Wažewski type approximation in the presence of state constraints. Our example also presents an occupation measure relaxation gap.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114036"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838860","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 : 2026-04-05Epub Date: 2025-12-22DOI: 10.1016/j.jde.2025.114042
Jianfeng Cheng , Xiaohui Wang , Wei Xiang
This paper is concerned with the well-posedness of the compressible subsonic jet flows issuing from a semi-infinitely long nozzle, the free streamline detaches smoothly from the nozzle wall, and the detachment is not known a priori. More specifically, given a semi-infinitely long de Laval type nozzle and an atmosphere pressure , there exists a critical value and an interval , such that for any incoming mass flux and the pressure difference , there exists a unique compressible subsonic jet flow and the detachment lies on the divergent part of the nozzle wall. Moreover, the detachment is continuous and strictly monotonic with respect to . Finally, we also establish the optimal -regularity of the free boundary at the detachment.
{"title":"Compressible subsonic jet flows with unprescribed detachment","authors":"Jianfeng Cheng , Xiaohui Wang , Wei Xiang","doi":"10.1016/j.jde.2025.114042","DOIUrl":"10.1016/j.jde.2025.114042","url":null,"abstract":"<div><div>This paper is concerned with the well-posedness of the compressible subsonic jet flows issuing from a semi-infinitely long nozzle, the free streamline detaches smoothly from the nozzle wall, and the detachment is not known a priori. More specifically, given a semi-infinitely long de Laval type nozzle and an atmosphere pressure <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>a</mi><mi>t</mi><mi>m</mi></mrow></msub><mo>></mo><mn>0</mn></math></span>, there exists a critical value <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi><mi>r</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> and an interval <span><math><mo>[</mo><munder><mrow><mi>p</mi></mrow><mo>_</mo></munder><mo>,</mo><mover><mrow><mi>p</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo></math></span>, such that for any incoming mass flux <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> and the pressure difference <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi><mi>i</mi><mi>f</mi></mrow></msub><mo>∈</mo><mo>[</mo><munder><mrow><mi>p</mi></mrow><mo>_</mo></munder><mo>,</mo><mover><mrow><mi>p</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo></math></span>, there exists a unique compressible subsonic jet flow and the detachment lies on the divergent part of the nozzle wall. Moreover, the detachment is continuous and strictly monotonic with respect to <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi><mi>i</mi><mi>f</mi></mrow></msub></math></span>. Finally, we also establish the optimal <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>-regularity of the free boundary at the detachment.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114042"},"PeriodicalIF":2.3,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838862","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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