Pub Date : 2026-01-13DOI: 10.1016/j.jde.2026.114101
Chengyu Wu, Jiaqing Yang
This paper is concerning the inverse conductive scattering of acoustic waves by a bounded inhomogeneous object with possibly embedded obstacles inside. A new uniqueness theorem is proved that the conductive object is uniquely determined by the fixed frequency far-field measurements, ignoring its contents. Meanwhile, the boundary informations of several related physical coefficients are also uniquely determined. The proof is mainly based on a detailed singularity analysis of solutions near the interface associated with a family of point sources or hypersingular point sources, which is deduced by the potential theory. Moreover, the other key ingredient in the proof is the well-posedness of the interior transmission problem with the conductivity boundary condition in the sense, where several sufficient conditions depending on the domain and physical coefficients are provided.
{"title":"On uniqueness of inverse conductive scattering problem with unknown embedded obstacles","authors":"Chengyu Wu, Jiaqing Yang","doi":"10.1016/j.jde.2026.114101","DOIUrl":"10.1016/j.jde.2026.114101","url":null,"abstract":"<div><div>This paper is concerning the inverse conductive scattering of acoustic waves by a bounded inhomogeneous object with possibly embedded obstacles inside. A new uniqueness theorem is proved that the conductive object is uniquely determined by the fixed frequency far-field measurements, ignoring its contents. Meanwhile, the boundary informations of several related physical coefficients are also uniquely determined. The proof is mainly based on a detailed singularity analysis of solutions near the interface associated with a family of point sources or hypersingular point sources, which is deduced by the potential theory. Moreover, the other key ingredient in the proof is the well-posedness of the interior transmission problem with the conductivity boundary condition in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> sense, where several sufficient conditions depending on the domain and physical coefficients are provided.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114101"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978282","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 : 2026-01-13DOI: 10.1016/j.jde.2026.114098
Chao Liu , Feng Dai , Bin Liu
<div><div>The Keller-Segel-Navier-Stokes system modeling coral fertilization <span><math><msubsup><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mi>∇</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><mi>κ</mi><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><mi>∇</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo><mi>∇</mi><mi>ϕ</mi></math></span>; <span><math><mi>∇</mi><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> is considered in a smoothly bounded convex domain. While global classical solutions <span><math><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo></math></span> exist for a
{"title":"The small-convection limit in a two-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization","authors":"Chao Liu , Feng Dai , Bin Liu","doi":"10.1016/j.jde.2026.114098","DOIUrl":"10.1016/j.jde.2026.114098","url":null,"abstract":"<div><div>The Keller-Segel-Navier-Stokes system modeling coral fertilization <span><math><msubsup><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mi>∇</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>; <span><math><msubsup><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>κ</mi></mrow></msubsup><mo>+</mo><mi>κ</mi><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>⋅</mo><mi>∇</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>−</mo><mi>∇</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>+</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo><mi>∇</mi><mi>ϕ</mi></math></span>; <span><math><mi>∇</mi><mo>⋅</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span> is considered in a smoothly bounded convex domain. While global classical solutions <span><math><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>c</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>m</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>,</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>)</mo></math></span> exist for a","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114098"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977523","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 : 2026-01-13DOI: 10.1016/j.jde.2026.114100
Huayan Su, Caibin Zeng
This paper investigates the nonlinear singular stochastic delay differential equation, extending beyond the multiplicative ergodic theorem. We establish a quasi-equivalent relationship between measurably contracting cone families and measurably dominated splittings in measurable fields of Banach spaces. Under an integrability condition, we derive a generalized Krein-Rutmann-type theorem for compact, injective linear cocycles on Banach spaces, without assuming cocycle compactness or integrability. The Lian-Wang index, rather than the Lyapunov norm, is employed to quantify contracting cone families and their eventual measurability. Leveraging smooth ergodic theory, we prove the existence of measurably dominated splitting in probability. Using the graph transform method, we further show that fields of Banach spaces admit measurably dominated splitting when cone invariance is satisfied. These results advance the understanding of nonlinear dynamics in stochastic systems with delays and singularities.
{"title":"Measurably dominated splitting of fields of Banach spaces: Beyond the multiplicative ergodic theorem","authors":"Huayan Su, Caibin Zeng","doi":"10.1016/j.jde.2026.114100","DOIUrl":"10.1016/j.jde.2026.114100","url":null,"abstract":"<div><div>This paper investigates the nonlinear singular stochastic delay differential equation, extending beyond the multiplicative ergodic theorem. We establish a quasi-equivalent relationship between measurably contracting cone families and measurably dominated splittings in measurable fields of Banach spaces. Under an integrability condition, we derive a generalized Krein-Rutmann-type theorem for compact, injective linear cocycles on Banach spaces, without assuming cocycle compactness or integrability. The Lian-Wang index, rather than the Lyapunov norm, is employed to quantify contracting cone families and their eventual measurability. Leveraging smooth ergodic theory, we prove the existence of measurably dominated splitting in probability. Using the graph transform method, we further show that fields of Banach spaces admit measurably dominated splitting when cone invariance is satisfied. These results advance the understanding of nonlinear dynamics in stochastic systems with delays and singularities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114100"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977525","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 : 2026-01-13DOI: 10.1016/j.jde.2025.114080
Peter Šepitka , Roman Šimon Hilscher , Vera M. Zeidan
In this paper we introduce a new approach suitable for studying the local oscillation properties of solutions to canonical systems defined on arbitrary hybrid time domains, also called general time scales. Such systems are known as symplectic or Hamiltonian systems on time scales. We define the notions of the local multiplicities of generalized left and right focal points for conjoined bases of the system and establish, among other results, a local version of the Sturm separation theorem. This result leads to a new concept in the oscillation theory on time scales, which we call the minimal multiplicity at the given point. We derive several properties of these minimal multiplicities with special focus on their zero value. Our analysis is based on the theory of comparative index and dual comparative index of two Lagrangian planes, which is introduced and applied for the first time in this paper to canonical systems on time scales. We also relate the local multiplicities of generalized focal points corresponding to two conjoined bases with the limit properties of the comparative index and the dual comparative index. This theory produces new results when also applied to matrix Jacobi systems arising in variational analysis over time scales or to second order Sturm–Liouville equations on time scales.
{"title":"Oscillation theory on hybrid time domains: Local oscillation properties","authors":"Peter Šepitka , Roman Šimon Hilscher , Vera M. Zeidan","doi":"10.1016/j.jde.2025.114080","DOIUrl":"10.1016/j.jde.2025.114080","url":null,"abstract":"<div><div>In this paper we introduce a new approach suitable for studying the local oscillation properties of solutions to canonical systems defined on arbitrary hybrid time domains, also called general time scales. Such systems are known as symplectic or Hamiltonian systems on time scales. We define the notions of the <em>local</em> multiplicities of generalized left and right focal points for conjoined bases of the system and establish, among other results, a local version of the Sturm separation theorem. This result leads to a new concept in the oscillation theory on time scales, which we call the <em>minimal multiplicity</em> at the given point. We derive several properties of these minimal multiplicities with special focus on their zero value. Our analysis is based on the theory of comparative index and dual comparative index of two Lagrangian planes, which is introduced and applied for the first time in this paper to canonical systems on <em>time scales</em>. We also relate the local multiplicities of generalized focal points corresponding to two conjoined bases with the limit properties of the comparative index and the dual comparative index. This theory produces new results when also applied to matrix Jacobi systems arising in variational analysis over time scales or to second order Sturm–Liouville equations on time scales.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 114080"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977784","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 : 2026-01-13DOI: 10.1016/j.jde.2025.114092
Wen-Xin Qin , Tong Zhou
The depinning force for the Frenkel-Kontorova chain is a critical value of the driving force F up to which there continue to be Birkhoff equilibria of rotation symbol ω and above which there are none. In this paper we investigate the modulus of continuity for the depinning force at rational rotation symbols and and obtain the estimate where C is a constant and denotes the underlying number associated to the rotation symbol ω. A similar conclusion for also holds true.
As an application, we give an open and dense result for , a threshold of driving force such that there exist stationary fronts for and traveling fronts for .
{"title":"Modulus of continuity for depinning force at rational rotation symbols and application","authors":"Wen-Xin Qin , Tong Zhou","doi":"10.1016/j.jde.2025.114092","DOIUrl":"10.1016/j.jde.2025.114092","url":null,"abstract":"<div><div>The depinning force for the Frenkel-Kontorova chain is a critical value <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>ω</mi><mo>)</mo></math></span> of the driving force <em>F</em> up to which there continue to be Birkhoff equilibria of rotation symbol <em>ω</em> and above which there are none. In this paper we investigate the modulus of continuity for the depinning force at rational rotation symbols <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo></math></span> and <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>−</mo></math></span> and obtain the estimate<span><span><span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo><mo>)</mo><mo>−</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>ω</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>C</mi><mo>|</mo><mi>q</mi><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mi>p</mi><mo>|</mo><mo>,</mo><mspace></mspace><mtext> for </mtext><mspace></mspace><mi>ω</mi><mo>></mo><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo><mo>,</mo></math></span></span></span> where <em>C</em> is a constant and <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> denotes the underlying number associated to the rotation symbol <em>ω</em>. A similar conclusion for <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>−</mo></math></span> also holds true.</div><div>As an application, we give an open and dense result for <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo><mo>></mo><mn>0</mn></math></span>, a threshold of driving force such that there exist stationary fronts for <span><math><mi>F</mi><mo>≤</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo></math></span> and traveling fronts for <span><math><mi>F</mi><mo>></mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114092"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977972","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 : 2026-01-13DOI: 10.1016/j.jde.2026.114095
Tuowei Chen , Qiangchang Ju
This paper is concerned with the two-dimensional full compressible Navier-Stokes equations between two infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.
{"title":"Global existence for full compressible Navier-Stokes equations around the Couette flow with a temperature gradient in an infinite channel","authors":"Tuowei Chen , Qiangchang Ju","doi":"10.1016/j.jde.2026.114095","DOIUrl":"10.1016/j.jde.2026.114095","url":null,"abstract":"<div><div>This paper is concerned with the two-dimensional full compressible Navier-Stokes equations between two infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114095"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977973","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 : 2026-01-12DOI: 10.1016/j.jde.2026.114102
Xiang Lv
We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs), which is very useful in a wide range of applications. As a consequence, we can show that the ω-limit sets of all pullback trajectories of semilinear/nonlinear stochastic differential equations (SDEs) with additive/multiplicative white noise are composed of nontrivial random equilibria. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.
{"title":"An abstract criterion on the existence and global stability of stationary solutions for random dynamical systems and its applications","authors":"Xiang Lv","doi":"10.1016/j.jde.2026.114102","DOIUrl":"10.1016/j.jde.2026.114102","url":null,"abstract":"<div><div>We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs), which is very useful in a wide range of applications. As a consequence, we can show that the <em>ω</em>-limit sets of all pullback trajectories of semilinear/nonlinear stochastic differential equations (SDEs) with additive/multiplicative white noise are composed of nontrivial random equilibria. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114102"},"PeriodicalIF":2.3,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977524","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 : 2026-01-12DOI: 10.1016/j.jde.2026.114097
Bin Qian , Min Wang , Ran Wang , Yimin Xiao
Consider the nonlinear stochastic heat equation where is a Gaussian noise which is white in time and fractional in space with Hurst parameter . The existence and uniqueness of the solutions to this equation were proved by Balan et al. [1] when is an affine function, and by Hu et al. [19] when σ is differentiable with Lipschitz derivative and . In both settings, the Hölder continuity of the solution has been proved by Balan et al. [2] and Hu et al. [19], respectively.
In this paper, we study the asymptotic behavior of the temporal increment for fixed and as , within the framework of [19]. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process , where is fixed.
考虑非线性随机热方程∂u(t,x)∂t=∂2u(t,x)∂x2+σ(u(t,x))W˙（t,x），t>0,x∈R，其中W˙是高斯噪声，在时间上是白的，在空间上是分数的，Hurst参数H∈(14,12)。当σ(u)=au+b是仿射函数时，Balan et al.[1]证明了该方程解的存在唯一性；当σ(0)=0时，σ可与Lipschitz导数微分时，Hu et al.[19]证明了该方程解的存在唯一性。在这两种情况下，分别由Balan et al.[2]和Hu et al.[19]证明了解的Hölder连续性。本文在[19]的框架下，研究了固定t≥0且x∈R为ε↓0时，时间增量u(t+ε,x)−u（t,x）的渐近性。作为应用，我们导出了迭代对数的Khinchin定律，迭代对数的Chung定律，以及时间过程{u(t,x)}t≥0的二次变分，其中x∈R是固定的。
{"title":"Temporal regularity for the nonlinear stochastic heat equation with spatially rough noise","authors":"Bin Qian , Min Wang , Ran Wang , Yimin Xiao","doi":"10.1016/j.jde.2026.114097","DOIUrl":"10.1016/j.jde.2026.114097","url":null,"abstract":"<div><div>Consider the nonlinear stochastic heat equation<span><span><span><math><mfrac><mrow><mo>∂</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>σ</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>)</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is a Gaussian noise which is white in time and fractional in space with Hurst parameter <span><math><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>. The existence and uniqueness of the solutions to this equation were proved by Balan et al. <span><span>[1]</span></span> when <span><math><mi>σ</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>u</mi><mo>+</mo><mi>b</mi></math></span> is an affine function, and by Hu et al. <span><span>[19]</span></span> when <em>σ</em> is differentiable with Lipschitz derivative and <span><math><mi>σ</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. In both settings, the Hölder continuity of the solution has been proved by Balan et al. <span><span>[2]</span></span> and Hu et al. <span><span>[19]</span></span>, respectively.</div><div>In this paper, we study the asymptotic behavior of the temporal increment <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>ε</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> for fixed <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> as <span><math><mi>ε</mi><mo>↓</mo><mn>0</mn></math></span>, within the framework of <span><span>[19]</span></span>. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process <span><math><msub><mrow><mo>{</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, where <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> is fixed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114097"},"PeriodicalIF":2.3,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980540","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 : 2026-01-09DOI: 10.1016/j.jde.2026.114096
Peng Shi , Yan-Xia Feng , Wan-Tong Li , Fei-Ying Yang
Recent studies indicate that in many epidemics, the strains (bacterial or viral) of disease-causing pathogens exhibit significant diversity, and human mobility patterns follow scale-free, nonlocal dynamics characterized by heavy-tailed distributions such as Lévy flights. To investigate the long-range geographical spread of multi-strain epidemics, this article proposes a multi-strain susceptible-infected-susceptible (SIS) model incorporating fractional diffusion. The central questions addressed in our study include the competitive exclusion and coexistence of multiple strains, as well as the influence of fractional powers and dispersal rates on the asymptotic behavior of equilibrium solutions. Our analysis demonstrates that: (i) the basic reproduction number acts as a threshold for disease extinction; (ii) the invasion number serves as a threshold for both the existence and stability of the coexistence equilibrium and the stability of single-strain endemic equilibria. Additionally, we examine the effect of home and hospital isolation measures on disease transmission.
{"title":"Spatiotemporal dynamics in a multi-strain epidemic model with fractional diffusion","authors":"Peng Shi , Yan-Xia Feng , Wan-Tong Li , Fei-Ying Yang","doi":"10.1016/j.jde.2026.114096","DOIUrl":"10.1016/j.jde.2026.114096","url":null,"abstract":"<div><div>Recent studies indicate that in many epidemics, the strains (bacterial or viral) of disease-causing pathogens exhibit significant diversity, and human mobility patterns follow scale-free, nonlocal dynamics characterized by heavy-tailed distributions such as Lévy flights. To investigate the long-range geographical spread of multi-strain epidemics, this article proposes a multi-strain susceptible-infected-susceptible (SIS) model incorporating fractional diffusion. The central questions addressed in our study include the competitive exclusion and coexistence of multiple strains, as well as the influence of fractional powers and dispersal rates on the asymptotic behavior of equilibrium solutions. Our analysis demonstrates that: (i) the basic reproduction number acts as a threshold for disease extinction; (ii) the invasion number serves as a threshold for both the existence and stability of the coexistence equilibrium and the stability of single-strain endemic equilibria. Additionally, we examine the effect of home and hospital isolation measures on disease transmission.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114096"},"PeriodicalIF":2.3,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924239","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 : 2026-01-08DOI: 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":"2026-01-08","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号