Pub Date : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114188
Grzegorz Świderski, Bartosz Trojan
We introduce a new class of Sturm–Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to 0. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.
{"title":"Sturm–Liouville operators with periodically modulated parameters. Part I: Regular case","authors":"Grzegorz Świderski, Bartosz Trojan","doi":"10.1016/j.jde.2026.114188","DOIUrl":"10.1016/j.jde.2026.114188","url":null,"abstract":"<div><div>We introduce a new class of Sturm–Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to 0. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114188"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192391","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 : 2026-05-25Epub Date: 2026-02-11DOI: 10.1016/j.jde.2026.114190
Huaizhi Cao, Hai-Yang Jin, Chenke Li
<div><div>This paper focuses on a spatiotemporal population-toxicant model with toxicant-taxis, subject to Dirichlet boundary conditions in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>χ</mi><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>w</mi><mo>)</mo><mo>+</mo><mi>u</mi><mo>(</mo><mi>r</mi><mo>−</mo><mi>a</mi><mi>u</mi><mo>)</mo><mo>−</mo><mi>m</mi><mi>u</mi><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>g</mi><mi>w</mi><mo>−</mo><mi>b</mi><mi>u</mi><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>∇</mi><mi>u</mi><mo>+</mo><mi>χ</mi><mi>u</mi><mi>∇</mi><mi>w</mi><mo>)</mo><mo>⋅</mo><mi>ν</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>w</mi><mo>=</mo><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>w</mi><mo>=</mo><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> represent the population density and toxicant concentration at position <em>x</em> and time <em>t</em>, respectively. Here, <span><math><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is a temporally and spatially heterogeneous input function describing the toxicant entering the environment through the boundary. Under some smoothness assumptions on <span><math><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>, we first establish the global existence of uniform-in-time bounded solutions via energy estimates in the case of <span><math><mi>n<
{"title":"Global dynamics of a toxicant-taxis model with Dirichlet boundary conditions","authors":"Huaizhi Cao, Hai-Yang Jin, Chenke Li","doi":"10.1016/j.jde.2026.114190","DOIUrl":"10.1016/j.jde.2026.114190","url":null,"abstract":"<div><div>This paper focuses on a spatiotemporal population-toxicant model with toxicant-taxis, subject to Dirichlet boundary conditions in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>χ</mi><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>w</mi><mo>)</mo><mo>+</mo><mi>u</mi><mo>(</mo><mi>r</mi><mo>−</mo><mi>a</mi><mi>u</mi><mo>)</mo><mo>−</mo><mi>m</mi><mi>u</mi><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>g</mi><mi>w</mi><mo>−</mo><mi>b</mi><mi>u</mi><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>∇</mi><mi>u</mi><mo>+</mo><mi>χ</mi><mi>u</mi><mi>∇</mi><mi>w</mi><mo>)</mo><mo>⋅</mo><mi>ν</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>w</mi><mo>=</mo><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>w</mi><mo>=</mo><mi>w</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> represent the population density and toxicant concentration at position <em>x</em> and time <em>t</em>, respectively. Here, <span><math><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is a temporally and spatially heterogeneous input function describing the toxicant entering the environment through the boundary. Under some smoothness assumptions on <span><math><mi>H</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>, we first establish the global existence of uniform-in-time bounded solutions via energy estimates in the case of <span><math><mi>n<","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114190"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192394","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 : 2026-05-25Epub Date: 2026-02-13DOI: 10.1016/j.jde.2026.114222
A. Della Corte , M. Farotti
We study chain-recurrence and chain-transitivity in compact dynamical systems without any regularity assumptions on the map. We prove that every compact system has a chain-recurrent point and a closed, invariant, chain-transitive subsystem. The proofs do not use the Axiom of Choice.
{"title":"Chains without regularity","authors":"A. Della Corte , M. Farotti","doi":"10.1016/j.jde.2026.114222","DOIUrl":"10.1016/j.jde.2026.114222","url":null,"abstract":"<div><div>We study chain-recurrence and chain-transitivity in compact dynamical systems without any regularity assumptions on the map. We prove that every compact system has a chain-recurrent point and a closed, invariant, chain-transitive subsystem. The proofs do not use the Axiom of Choice.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114222"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192387","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114198
Peng Qu , Jiahui Wang
Entropy solutions with periodic initial data are considered for the quasi-one-dimensional isentropic Euler system. An approximate solution sequence is constructed using a fractional Glimm scheme, and then wave interactions are carefully analyzed. Energy behavior of the system is studied as an extra approximate conservation law in order to overcome the difficulty caused by the non–conservation property. After that, the method of approximate conservation laws and approximate characteristics is applied to analyze the decay of the system, which gives the uniform total variation bounds and thus the global existence.
{"title":"Periodic entropy weak solutions for quasi–one–dimensional isentropic Euler flows with periodic initial data","authors":"Peng Qu , Jiahui Wang","doi":"10.1016/j.jde.2026.114198","DOIUrl":"10.1016/j.jde.2026.114198","url":null,"abstract":"<div><div>Entropy solutions with periodic initial data are considered for the quasi-one-dimensional isentropic Euler system. An approximate solution sequence is constructed using a fractional Glimm scheme, and then wave interactions are carefully analyzed. Energy behavior of the system is studied as an extra approximate conservation law in order to overcome the difficulty caused by the non–conservation property. After that, the method of approximate conservation laws and approximate characteristics is applied to analyze the decay of the system, which gives the uniform total variation bounds and thus the global existence.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114198"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192389","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114200
Feiying Yan , Xianguo Geng , Jingru Geng
In this paper, we propose a new complex nonlinear model and derive its Lax pair, where β is an arbitrary real constant. Using the Riemann-Hilbert method and -steepest descent method, we obtain soliton resolution conjecture and the long-time asymptotics for the integrable complex nonlinear model in the presence of discrete spectrum. To be more specific, resorting to the spectral analysis of Lax pair, the solution of the Cauchy problem of the integrable complex nonlinear model is characterized by the solution of the derived Riemann-Hilbert problem in the new scale . Based on the categorization of , the long-time asymptotic expansion of the solution in space-time soliton regions is obtained by using a series of contour deformations. We finally obtain the soliton resolution and long-time asymptotics of the integrable complex nonlinear model with the aid of the -steepest descent method, in which the leading term is characterized by an -soliton on the discrete spectrum, the second term comes from the soliton radiation interactions on the continuum spectrum, and the error term is generated by the corresponding -problem. The results also indicate that the N-soliton solutions for the integrable complex nonlinear model are asymptotically stable.
{"title":"The Cauchy problem of an integrable complex nonlinear model with weighted Sobolev initial data: soliton resolution and asymptotic stability of N-solitons","authors":"Feiying Yan , Xianguo Geng , Jingru Geng","doi":"10.1016/j.jde.2026.114200","DOIUrl":"10.1016/j.jde.2026.114200","url":null,"abstract":"<div><div>In this paper, we propose a new complex nonlinear model<span><span><span><math><mi>i</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mo>(</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>−</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo>−</mo><mn>2</mn><mi>β</mi><mi>u</mi><mo>,</mo></math></span></span></span> and derive its Lax pair, where <em>β</em> is an arbitrary real constant. Using the Riemann-Hilbert method and <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method, we obtain soliton resolution conjecture and the long-time asymptotics for the integrable complex nonlinear model in the presence of discrete spectrum. To be more specific, resorting to the spectral analysis of Lax pair, the solution of the Cauchy problem of the integrable complex nonlinear model is characterized by the solution of the derived Riemann-Hilbert problem in the new scale <span><math><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mi>t</mi><mo>)</mo></math></span>. Based on the categorization of <span><math><mover><mrow><mi>ξ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>=</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>/</mo><mi>t</mi></math></span>, the long-time asymptotic expansion of the solution <span><math><mi>u</mi><mo>(</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mi>t</mi><mo>)</mo></math></span> in space-time soliton regions is obtained by using a series of contour deformations. We finally obtain the soliton resolution and long-time asymptotics of the integrable complex nonlinear model with the aid of the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method, in which the leading term is characterized by an <span><math><mi>N</mi><mo>(</mo><mi>Λ</mi><mo>)</mo></math></span>-soliton on the discrete spectrum, the second term comes from the soliton radiation interactions on the continuum spectrum, and the error term <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> is generated by the corresponding <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-problem. The results also indicate that the <em>N</em>-soliton solutions for the integrable complex nonlinear model are asymptotically stable.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114200"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192396","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 : 2026-05-25Epub Date: 2026-02-13DOI: 10.1016/j.jde.2026.114224
Yuxia Guo, Tingfeng Yuan
We consider the following parabolic degenerate equation:(0.1) where and . By using backward similarity variables and energy estimate method developed in [25], we prove a Liouville type theorem for non-negative classical solutions of (0.1) under some restrictions on a and p. In particular, for the specific case , we obtain the result for the full subcritical range () under an additional monotonicity assumption.
{"title":"Liouville-type theorem for parabolic degenerate equation","authors":"Yuxia Guo, Tingfeng Yuan","doi":"10.1016/j.jde.2026.114224","DOIUrl":"10.1016/j.jde.2026.114224","url":null,"abstract":"<div><div>We consider the following parabolic degenerate equation:<span><span><span>(0.1)</span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>a</mi></mrow></msub><mi>u</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>×</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>+</mo><mi>a</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>y</mi></mrow></msub><mo>+</mo><mi>y</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>y</mi><mi>y</mi></mrow></msub></math></span> and <span><math><mi>a</mi><mo>≥</mo><mn>1</mn></math></span>. By using backward similarity variables and energy estimate method developed in <span><span>[25]</span></span>, we prove a Liouville type theorem for non-negative classical solutions of <span><span>(0.1)</span></span> under some restrictions on <em>a</em> and <em>p</em>. In particular, for the specific case <span><math><mi>a</mi><mo>=</mo><mn>1</mn></math></span>, we obtain the result for the full subcritical range (<span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>N</mi></mrow></mfrac></math></span>) under an additional monotonicity assumption.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114224"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192385","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114189
João Henrique Andrade , Dario Corona , Stefano Nardulli , Paolo Piccione , Raoní Ponciano
We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the volume constraint is sufficiently small. This system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the regularization parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, and solutions concentrate in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters, which is incomplete for a larger number of phases. To address this issue, we employ the “volume-fixing variations” approach, enabling us to establish results for any number of phases. This offers more profound insights into phase separation phenomena on manifolds with arbitrary geometry.
{"title":"From bubbles to clusters: Multiple solutions to the Allen–Cahn system","authors":"João Henrique Andrade , Dario Corona , Stefano Nardulli , Paolo Piccione , Raoní Ponciano","doi":"10.1016/j.jde.2026.114189","DOIUrl":"10.1016/j.jde.2026.114189","url":null,"abstract":"<div><div>We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the volume constraint is sufficiently small. This system naturally arises in phase separation models, where solutions represent the distribution of distinct phases in a multi-component mixture. As the regularization parameter approaches zero, the system's energy approximates the multi-isoperimetric profile, and solutions concentrate in regions resembling isoperimetric clusters. For two or three phases, these results rely on classifying isoperimetric clusters, which is incomplete for a larger number of phases. To address this issue, we employ the “volume-fixing variations” approach, enabling us to establish results for any number of phases. This offers more profound insights into phase separation phenomena on manifolds with arbitrary geometry.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114189"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192395","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114191
Jie Fan , HongJie Ju , YanNan Liu
In this paper, we study the Gaussian dual Minkowski problem. By using flow method, we obtain an existence result of even solutions for smooth even measures when . If , we show that under some restrictions on , the only smooth, uniformly convex and origin-centered solution is the standard ball.
{"title":"Existence and uniqueness of solutions to the even Lp Gaussian dual Minkowski problem","authors":"Jie Fan , HongJie Ju , YanNan Liu","doi":"10.1016/j.jde.2026.114191","DOIUrl":"10.1016/j.jde.2026.114191","url":null,"abstract":"<div><div>In this paper, we study the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> Gaussian dual Minkowski problem. By using flow method, we obtain an existence result of even solutions for smooth even measures when <span><math><mi>p</mi><mi>q</mi><mo>></mo><mn>0</mn></math></span>. If <span><math><mi>f</mi><mo>≡</mo><mn>1</mn></math></span>, we show that under some restrictions on <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span>, the only smooth, uniformly convex and origin-centered solution is the standard ball.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114191"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192392","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114199
Hyunjin In , Dong-ha Kim , Junha Kim
We investigate the asymptotic stability of a tropical climate model posed on , with temperature-dependent diffusion in the barotropic mode u and linear damping in the first baroclinic mode v. We consider two distinct cases for the barotropic component: one with linear damping and one without. For both cases, we prove the small data global existence of smooth solutions. Furthermore, we establish sharp temporal decay estimates for solutions in arbitrary Sobolev norms , .
{"title":"Asymptotic stability of the 2D temperature-dependent tropical climate model with the sharp decay rates","authors":"Hyunjin In , Dong-ha Kim , Junha Kim","doi":"10.1016/j.jde.2026.114199","DOIUrl":"10.1016/j.jde.2026.114199","url":null,"abstract":"<div><div>We investigate the asymptotic stability of a tropical climate model posed on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, with temperature-dependent diffusion in the barotropic mode <em>u</em> and linear damping in the first baroclinic mode <em>v</em>. We consider two distinct cases for the barotropic component: one with linear damping and one without. For both cases, we prove the small data global existence of smooth solutions. Furthermore, we establish sharp temporal decay estimates for solutions in arbitrary Sobolev norms <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114199"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192386","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 : 2026-05-25Epub Date: 2026-02-10DOI: 10.1016/j.jde.2026.114193
Nicolò Drago , Sonia Mazzucchi , Andrea Pinamonti
This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schrödinger equations on the Heisenberg group . Using semigroup approximation techniques, we obtain analytically tractable and numerically implementable representations of fundamental solutions. In particular, we establish a new connection between the heat equation and Brownian motion on and provide a rigorous realization of the Feynman path integral for the Schrödinger equation. The study highlights the challenges posed by the noncommutative structure of the Heisenberg group and opens new directions for PDEs on sub-Riemannian manifolds.
{"title":"Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group","authors":"Nicolò Drago , Sonia Mazzucchi , Andrea Pinamonti","doi":"10.1016/j.jde.2026.114193","DOIUrl":"10.1016/j.jde.2026.114193","url":null,"abstract":"<div><div>This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schrödinger equations on the Heisenberg group <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. Using semigroup approximation techniques, we obtain analytically tractable and numerically implementable representations of fundamental solutions. In particular, we establish a new connection between the heat equation and Brownian motion on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> and provide a rigorous realization of the Feynman path integral for the Schrödinger equation. The study highlights the challenges posed by the noncommutative structure of the Heisenberg group and opens new directions for PDEs on sub-Riemannian manifolds.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"464 ","pages":"Article 114193"},"PeriodicalIF":2.3,"publicationDate":"2026-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192390","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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