Pub Date : 2025-12-05DOI: 10.1016/j.jde.2025.113997
Sungwon Cho , Junyuan Fang , Tuoc Phan
This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate through a weight belonging to the class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is crucially applied to suitable barrier functions to control the solutions. As corollaries, Hölder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.
{"title":"Harnack inequality for singular or degenerate parabolic equations in non-divergence form","authors":"Sungwon Cho , Junyuan Fang , Tuoc Phan","doi":"10.1016/j.jde.2025.113997","DOIUrl":"10.1016/j.jde.2025.113997","url":null,"abstract":"<div><div>This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate through a weight belonging to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msub></math></span> class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is crucially applied to suitable barrier functions to control the solutions. As corollaries, Hölder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113997"},"PeriodicalIF":2.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693187","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 : 2025-12-05DOI: 10.1016/j.jde.2025.113993
The Anh Bui , Xuan Thinh Duong , Guorong Hu , Ji Li , Brett D. Wick
Let Δ be the Laplace–Beltrami operator acting on a non-doubling manifold with two ends . In this paper, we will prove the following estimate Hence, by interpolation, for and , These can be viewed as sharp estimates for Schrödinger flows associated with the Laplace–Beltrami operator Δ. We note that these results also hold for more general second order differential operator whose heat kernel satisfies the same upper bound as the Laplace–Beltrami operator Δ, such as the Schrödinger operator with non-negative potential V.
{"title":"Sharp estimates for Schrödinger groups on non-doubling manifolds with ends","authors":"The Anh Bui , Xuan Thinh Duong , Guorong Hu , Ji Li , Brett D. Wick","doi":"10.1016/j.jde.2025.113993","DOIUrl":"10.1016/j.jde.2025.113993","url":null,"abstract":"<div><div>Let Δ be the Laplace–Beltrami operator acting on a non-doubling manifold with two ends <span><math><mi>M</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>♯</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>m</mi><mo>></mo><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. In this paper, we will prove the following estimate<span><span><span><math><msub><mrow><mo>‖</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>m</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>τ</mi><mi>Δ</mi></mrow></msup><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>≤</mo><mi>C</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>τ</mi><mo>|</mo><mo>)</mo></mrow><mrow><mi>m</mi><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>,</mo><mspace></mspace><mo>∀</mo><mi>τ</mi><mo>∈</mo><mi>R</mi><mo>.</mo></math></span></span></span> Hence, by interpolation, for <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> and <span><math><mi>s</mi><mo>=</mo><mi>m</mi><mo>|</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>p</mi><mo>|</mo></math></span>,<span><span><span><math><msub><mrow><mo>‖</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>τ</mi><mi>Δ</mi></mrow></msup><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>≤</mo><mi>C</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>τ</mi><mo>|</mo><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mo>‖</mo><mi>f</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msub><mo>,</mo><mspace></mspace><mo>∀</mo><mi>τ</mi><mo>∈</mo><mi>R</mi><mo>.</mo></math></span></span></span> These can be viewed as sharp estimates for Schrödinger flows associated with the Laplace–Beltrami operator Δ. We note that these results also hold for more general second order differential operator <span><math><mi>L</mi></math></span> whose heat kernel satisfies the same upper bound as the Laplace–Beltrami operator Δ, such as the Schrödinger operator <span><math><mi>L</mi><mo>=</mo><mi>Δ</mi><mo>+</mo><mi>V</mi></math></span> with non-negative potential <em>V</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113993"},"PeriodicalIF":2.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693104","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 : 2025-12-05DOI: 10.1016/j.jde.2025.113995
Guo Lin , Wenqing Xu, Xiying Yang
This article mainly studies the propagation dynamics of non-cooperative reaction-diffusion systems, which can be regarded as SIRI epidemic models incorporating relapse and a bilinear incidence rate. Assuming that the target population evolves in a front-like pattern, we analyze the initial value problem and traveling wave solutions to characterize the spread or extinction of the disease. For the initial value problem, its spreading properties under various scenarios are presented. Regarding to traveling wave solutions, we investigate two distinct asymptotic boundary conditions and establish the existence and nonexistence of nontrivial traveling wave solutions. Notably, a special wave profile set is constructed to confirm the existence of persistent traveling wave solutions, which reflects the relationship between the two branches of traveling wave solutions. Finally, we extend our analytical framework to investigate the propagation dynamics of non-monotonic delayed equations in shifting environments, as well as SIRI models with general incidence functions.
{"title":"Propagation dynamics of non-cooperative systems and applications to delayed equations","authors":"Guo Lin , Wenqing Xu, Xiying Yang","doi":"10.1016/j.jde.2025.113995","DOIUrl":"10.1016/j.jde.2025.113995","url":null,"abstract":"<div><div>This article mainly studies the propagation dynamics of non-cooperative reaction-diffusion systems, which can be regarded as SIRI epidemic models incorporating relapse and a bilinear incidence rate. Assuming that the target population evolves in a front-like pattern, we analyze the initial value problem and traveling wave solutions to characterize the spread or extinction of the disease. For the initial value problem, its spreading properties under various scenarios are presented. Regarding to traveling wave solutions, we investigate two distinct asymptotic boundary conditions and establish the existence and nonexistence of nontrivial traveling wave solutions. Notably, a special wave profile set is constructed to confirm the existence of persistent traveling wave solutions, which reflects the relationship between the two branches of traveling wave solutions. Finally, we extend our analytical framework to investigate the propagation dynamics of non-monotonic delayed equations in shifting environments, as well as SIRI models with general incidence functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113995"},"PeriodicalIF":2.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693190","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 : 2025-12-05DOI: 10.1016/j.jde.2025.113987
Igor Kukavica , Fanhui Xu
We address the global-in-time existence and pathwise uniqueness of solutions for the three-dimensional incompressible stochastic Navier-Stokes equations with multiplicative noise. Under natural smallness conditions on the noise, we prove an almost sure global existence result for small initial data in . Specifically, we show that sufficiently small data yields a pathwise unique strong solution that remains small in and is global-in-time on a set of probability close to 1, with this probability increasing as the initial norm decreases. Moreover, the solution decays exponentially with large probability.
{"title":"On the almost global existence of the stochastic Navier-Stokes equations in L3 with small data","authors":"Igor Kukavica , Fanhui Xu","doi":"10.1016/j.jde.2025.113987","DOIUrl":"10.1016/j.jde.2025.113987","url":null,"abstract":"<div><div>We address the global-in-time existence and pathwise uniqueness of solutions for the three-dimensional incompressible stochastic Navier-Stokes equations with multiplicative noise. Under natural smallness conditions on the noise, we prove an almost sure global existence result for small initial data in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Specifically, we show that sufficiently small <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> data yields a pathwise unique strong solution that remains small in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and is global-in-time on a set of probability close to 1, with this probability increasing as the initial <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> norm decreases. Moreover, the solution decays exponentially with large probability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113987"},"PeriodicalIF":2.3,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693092","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 : 2025-12-04DOI: 10.1016/j.jde.2025.113994
Jun Wu , Boling Guo , Zhong Wang
We investigate the asymptotic stability of a family of solitons in the energy space of the Hirota-Satsuma system. Specifically, we demonstrate that solutions in the neighborhood of these solitons converge to limiting objects, which, due to rigidity results, must also be solitons. In addition, we present a new scenario that each component of the solutions possesses distinct asymptotic velocities, which is quite different with respect to the case of scalar generalized Korteweg-de Vries (gKdV) equation. Our proof strategy is inspired by prior works on the gKdV equations [24], [26], and it incorporates the Liouville property for -compact solutions near the solitons.
{"title":"Asymptotic stability of solitons for the Hirota-Satsuma system","authors":"Jun Wu , Boling Guo , Zhong Wang","doi":"10.1016/j.jde.2025.113994","DOIUrl":"10.1016/j.jde.2025.113994","url":null,"abstract":"<div><div>We investigate the asymptotic stability of a family of solitons in the energy space of the Hirota-Satsuma system. Specifically, we demonstrate that solutions in the neighborhood of these solitons converge to limiting objects, which, due to rigidity results, must also be solitons. In addition, we present a new scenario that each component of the solutions possesses distinct asymptotic velocities, which is quite different with respect to the case of scalar generalized Korteweg-de Vries (gKdV) equation. Our proof strategy is inspired by prior works on the gKdV equations <span><span>[24]</span></span>, <span><span>[26]</span></span>, and it incorporates the Liouville property for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-compact solutions near the solitons.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113994"},"PeriodicalIF":2.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658762","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 : 2025-12-04DOI: 10.1016/j.jde.2025.114005
J.M. Burgos
We prove that total instability is a generic phenomenon in the real analytic class of electromagnetic Lagrangian systems under a weak magnetism hypothesis. The main object in the proof is an adaptation of the McGehee blowup for these systems. Together with this result, new criteria for total instability are introduced for both generic and non-generic cases.
{"title":"McGehee blowup for Lagrangian systems and instability of equilibria","authors":"J.M. Burgos","doi":"10.1016/j.jde.2025.114005","DOIUrl":"10.1016/j.jde.2025.114005","url":null,"abstract":"<div><div>We prove that total instability is a generic phenomenon in the real analytic class of electromagnetic Lagrangian systems under a weak magnetism hypothesis. The main object in the proof is an adaptation of the McGehee blowup for these systems. Together with this result, new criteria for total instability are introduced for both generic and non-generic cases.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"454 ","pages":"Article 114005"},"PeriodicalIF":2.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145681955","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 : 2025-12-04DOI: 10.1016/j.jde.2025.113884
Nick Lindemulder, Emiel Lorist , Floris B. Roodenburg , Mark C. Veraar
We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded -functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded -domains with , revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable.
{"title":"Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains","authors":"Nick Lindemulder, Emiel Lorist , Floris B. Roodenburg , Mark C. Veraar","doi":"10.1016/j.jde.2025.113884","DOIUrl":"10.1016/j.jde.2025.113884","url":null,"abstract":"<div><div>We study the Laplace operator on domains subject to Dirichlet or Neumann boundary conditions. We show that these operators admit a bounded <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-functional calculus on weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Our analysis applies to bounded <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>λ</mi></mrow></msup></math></span>-domains with <span><math><mi>λ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, revealing a crucial trade-off: lower domain regularity can be compensated by enlarging the weight exponent. As a primary consequence, we establish maximal regularity for the corresponding heat equation. This extends the well-posedness theory for parabolic equations to domains with minimal smoothness, where classical methods are inapplicable.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"454 ","pages":"Article 113884"},"PeriodicalIF":2.3,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145681956","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 : 2025-12-03DOI: 10.1016/j.jde.2025.113992
Zhikai Huang , Yongkai Liao , Huijiang Zhao
We study the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional viscous radiative and reactive gas model in this paper. Unlike the ideal gases, it is shown in [11] that the pressure may not be a convex function with respect to the specific volume and the specific entropy, which makes it difficult to establish the basic energy estimates for the system. By dealing with the nonlinear radiative terms cleverly, we conquer the above difficulty and establish the nonlinear stability of rarefaction waves for the system under large initial perturbation with general radiation constant. The results in this paper improve upon that obtained in [11] by removing the smallness assumption imposed in the radiation constant and broadening the range of the parameters .
{"title":"Global stability of rarefaction waves for a viscous radiative and reactive gas","authors":"Zhikai Huang , Yongkai Liao , Huijiang Zhao","doi":"10.1016/j.jde.2025.113992","DOIUrl":"10.1016/j.jde.2025.113992","url":null,"abstract":"<div><div>We study the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem of a one-dimensional viscous radiative and reactive gas model in this paper. Unlike the ideal gases, it is shown in [11] that the pressure may not be a convex function with respect to the specific volume and the specific entropy, which makes it difficult to establish the basic energy estimates for the system. By dealing with the nonlinear radiative terms cleverly, we conquer the above difficulty and establish the nonlinear stability of rarefaction waves for the system under large initial perturbation with general radiation constant. The results in this paper improve upon that obtained in [11] by removing the smallness assumption imposed in the radiation constant and broadening the range of the parameters <span><math><mo>(</mo><mi>b</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 113992"},"PeriodicalIF":2.3,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658700","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 : 2025-12-02DOI: 10.1016/j.jde.2025.113967
Ryo Ito , Hirokazu Ninomiya
This paper investigates unbounded traveling wave solutions of one-dimensional reaction-diffusion equations, focusing on two key aspects: their existence and the relationship between the theories of bounded and unbounded traveling wave solutions. We establish the existence of a threshold speed, referred to as the minimal speed, which distinguishes the existence and non-existence of unbounded waves under mild technical assumptions on the nonlinearity, including the bistable case. Notably, we propose a min-max type characterization of wave speeds, using a traveling semi-wave solution derived by truncating unbounded waves. We determine the speeds of the traveling wave solutions of several reaction diffusion equations and illustrate the connection between bounded and unbounded traveling wave solutions.
{"title":"Theory of unbounded traveling wave solutions of reaction-diffusion equations: Existence and connections to classical theory","authors":"Ryo Ito , Hirokazu Ninomiya","doi":"10.1016/j.jde.2025.113967","DOIUrl":"10.1016/j.jde.2025.113967","url":null,"abstract":"<div><div>This paper investigates unbounded traveling wave solutions of one-dimensional reaction-diffusion equations, focusing on two key aspects: their existence and the relationship between the theories of bounded and unbounded traveling wave solutions. We establish the existence of a threshold speed, referred to as the minimal speed, which distinguishes the existence and non-existence of unbounded waves under mild technical assumptions on the nonlinearity, including the bistable case. Notably, we propose a min-max type characterization of wave speeds, using a traveling semi-wave solution derived by truncating unbounded waves. We determine the speeds of the traveling wave solutions of several reaction diffusion equations and illustrate the connection between bounded and unbounded traveling wave solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"456 ","pages":"Article 113967"},"PeriodicalIF":2.3,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682728","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 : 2025-12-02DOI: 10.1016/j.jde.2025.113990
Feng Dai , Bin Liu
<div><div>The paper is concerned with the Keller-Segel-Stokes system with a rapidly diffusing indirect signal<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>n</mi><mo>=</mo><mi>Δ</mi><mi>n</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo><mi>∇</mi><mi>v</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>v</mi><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>w</mi><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>n</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>∇</mi><mi>P</mi><mo>+</mo><mi>n</mi><mi>∇</mi><mi>ϕ</mi><mo>,</mo><mspace></mspace><mi>∇</mi><mo>⋅</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace></mspace><mo>(</mo><mo>⋆</mo><mo>)</mo></math></span></span></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with smooth boundary, where <span><math><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>, and the chemotactic sensitivity function <span><math><mi>S</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span> satisfies <span><math><mn>0</mn><mo>≤</mo><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> with some <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mi>R</mi></math></span>. It is proved that for all suitably regular initial data, a corresponding no-flux/no-flux/no-flux/Dirichlet initial boundary value problem of <span><math><mo>(</mo><mo>⋆</mo><mo>)</mo></math></span> possesses a globally bounded classical solution for arbitrarily weak saturation effect on chemotactic sensitivity (i.e., <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>). Moreover, this solution is shown to exponentially stabilize toward the corresponding spatially homogeneous equil
{"title":"Boundedness and stabilization in a three-dimensional Keller-Segel-Stokes system with a rapidly diffusing indirect signal","authors":"Feng Dai , Bin Liu","doi":"10.1016/j.jde.2025.113990","DOIUrl":"10.1016/j.jde.2025.113990","url":null,"abstract":"<div><div>The paper is concerned with the Keller-Segel-Stokes system with a rapidly diffusing indirect signal<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>n</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>n</mi><mo>=</mo><mi>Δ</mi><mi>n</mi><mo>−</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo><mi>∇</mi><mi>v</mi><mo>)</mo><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>v</mi><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>⋅</mo><mi>∇</mi><mi>w</mi><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>n</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mi>∇</mi><mi>P</mi><mo>+</mo><mi>n</mi><mi>∇</mi><mi>ϕ</mi><mo>,</mo><mspace></mspace><mi>∇</mi><mo>⋅</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mtd></mtr></mtable></mrow><mspace></mspace><mo>(</mo><mo>⋆</mo><mo>)</mo></math></span></span></span> in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with smooth boundary, where <span><math><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>, and the chemotactic sensitivity function <span><math><mi>S</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span> satisfies <span><math><mn>0</mn><mo>≤</mo><mi>S</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>S</mi></mrow></msub><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> with some <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>></mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mi>R</mi></math></span>. It is proved that for all suitably regular initial data, a corresponding no-flux/no-flux/no-flux/Dirichlet initial boundary value problem of <span><math><mo>(</mo><mo>⋆</mo><mo>)</mo></math></span> possesses a globally bounded classical solution for arbitrarily weak saturation effect on chemotactic sensitivity (i.e., <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>). Moreover, this solution is shown to exponentially stabilize toward the corresponding spatially homogeneous equil","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"456 ","pages":"Article 113990"},"PeriodicalIF":2.3,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145681043","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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