Pub Date : 2025-12-15DOI: 10.1016/j.nonrwa.2025.104577
Manjun Ma, Kaili Wang, Dan Li
This work is concerned with a nonlinear and non-monotonic reaction-diffusion system that models the dynamics of bacterial colonies with density-suppressed motility. We first establish the existence of global solutions and the attractivity of the uniform coexsitence state in a moving coordinate frame. Traveling waves are then transformed into fixed points of a mapping associated with an auxiliary system. By constructing upper and lower solutions, we next establish an invariant function space for this mapping. By using Schauder’s fixed point theorem, we derive implicit conditions for the existence of traveling waves. Through developing innovative analytical techniques, we further obtain explicit conditions that are corroborated by numerical computation and simulations of the considered bacterial colony model.
{"title":"Traveling waves in a bacterial colony model","authors":"Manjun Ma, Kaili Wang, Dan Li","doi":"10.1016/j.nonrwa.2025.104577","DOIUrl":"10.1016/j.nonrwa.2025.104577","url":null,"abstract":"<div><div>This work is concerned with a nonlinear and non-monotonic reaction-diffusion system that models the dynamics of bacterial colonies with density-suppressed motility. We first establish the existence of global solutions and the attractivity of the uniform coexsitence state in a moving coordinate frame. Traveling waves are then transformed into fixed points of a mapping associated with an auxiliary system. By constructing upper and lower solutions, we next establish an invariant function space for this mapping. By using Schauder’s fixed point theorem, we derive implicit conditions for the existence of traveling waves. Through developing innovative analytical techniques, we further obtain explicit conditions that are corroborated by numerical computation and simulations of the considered bacterial colony model.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104577"},"PeriodicalIF":1.8,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-13DOI: 10.1016/j.nonrwa.2025.104574
Weidong Qin, Yunxian Dai, Doudou Lou
This paper investigates a delayed predator-prey model incorporating fear effects, prey refuge, Crowley-Martin type functional response, and cross-diffusion. First, we analyze the existence and stability of the positive equilibrium of the non-delay model. Then, we investigate the conditions for the occurrence of Turing instability in the delayed model. The amplitude equation is derived using the multiple-scale perturbation method, revealing the relationship between pattern selection and system parameters. Meanwhile, some numerical simulations are conducted to validate the accuracy of the theoretical analysis. The results demonstrate that varying control parameters can induce diverse patterns, including spots, stripes, and mixed patterns. Additionally, we find that the fear response delay affects the stabilization time of patterns, and as the delay increases, the patterns gradually become unstable. This study highlights the impact of the fear response delay on the stability and pattern formation in predator-prey systems, providing theoretical insights into the complexity of population dynamics.
{"title":"Pattern dynamics in a reaction-diffusion predator-prey model with fear response delay","authors":"Weidong Qin, Yunxian Dai, Doudou Lou","doi":"10.1016/j.nonrwa.2025.104574","DOIUrl":"10.1016/j.nonrwa.2025.104574","url":null,"abstract":"<div><div>This paper investigates a delayed predator-prey model incorporating fear effects, prey refuge, Crowley-Martin type functional response, and cross-diffusion. First, we analyze the existence and stability of the positive equilibrium of the non-delay model. Then, we investigate the conditions for the occurrence of Turing instability in the delayed model. The amplitude equation is derived using the multiple-scale perturbation method, revealing the relationship between pattern selection and system parameters. Meanwhile, some numerical simulations are conducted to validate the accuracy of the theoretical analysis. The results demonstrate that varying control parameters can induce diverse patterns, including spots, stripes, and mixed patterns. Additionally, we find that the fear response delay affects the stabilization time of patterns, and as the delay increases, the patterns gradually become unstable. This study highlights the impact of the fear response delay on the stability and pattern formation in predator-prey systems, providing theoretical insights into the complexity of population dynamics.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104574"},"PeriodicalIF":1.8,"publicationDate":"2025-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-12DOI: 10.1016/j.nonrwa.2025.104575
Xinyu Bo, Qi An, Wenjun Liu, Guangying Lv
Considering the increasing level of international environmental pollution, especially the discharge of nuclear effluents into the oceans, we consider in this paper, the dynamics of a predator-prey model in toxic environments. The concentration of toxins is no longer constant, but is influenced by time and location, and it will interact with predator-prey systems, thereby affecting the dynamic behavior of the entire ecosystem. The boundedness and positive definiteness of the model are obtained by using the comparison principle and the maximum principle. Afterwards, the threshold condition of toxicant concentration for the stability of the steady state solutions and the rate of convergence of the solutions are obtained by using the matrix positive definiteness, Schauder’s theorem and LaSalle’s invariance principle. Finally, numerical examples verify our results.
{"title":"A predator-prey model with poison dependent diffusion","authors":"Xinyu Bo, Qi An, Wenjun Liu, Guangying Lv","doi":"10.1016/j.nonrwa.2025.104575","DOIUrl":"10.1016/j.nonrwa.2025.104575","url":null,"abstract":"<div><div>Considering the increasing level of international environmental pollution, especially the discharge of nuclear effluents into the oceans, we consider in this paper, the dynamics of a predator-prey model in toxic environments. The concentration of toxins is no longer constant, but is influenced by time and location, and it will interact with predator-prey systems, thereby affecting the dynamic behavior of the entire ecosystem. The boundedness and positive definiteness of the model are obtained by using the comparison principle and the maximum principle. Afterwards, the threshold condition of toxicant concentration for the stability of the steady state solutions and the rate of convergence of the solutions are obtained by using the matrix positive definiteness, Schauder’s theorem and LaSalle’s invariance principle. Finally, numerical examples verify our results.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"91 ","pages":"Article 104575"},"PeriodicalIF":1.8,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145738910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-11DOI: 10.1016/j.nonrwa.2025.104566
Zihao Zhang
This paper concerns subsonic Euler flows in a two-dimensional finitely long slightly curved nozzle under the vertical gravity. Concerning the effect of the vertical gravity, we first establish the existence of subsonic shear flows in the flat nozzle. We then investigate the structural stability of these background subsonic flows under small perturbations of suitable boundary conditions on the entrance and exit and the upper and lower nozzle walls. It can be formulated as a nonlinear boundary value problem for a hyperbolic-elliptic mixed system. The main difficulty is that all the physical quantities are coupled with each other due to the existence of the vertical gravity. The approach is based on the Lagrangian transformation to straighten the streamline and the deformation-curl decomposition to deal with the hyperbolic and elliptic modes in the subsonic region. The key ingredient of the analysis is to solve the associated linearized elliptic boundary value problem with mixed boundary conditions in a weighted Hölder space.
{"title":"Subsonic Euler flows with gravity in a two-dimensional finitely long curved nozzle","authors":"Zihao Zhang","doi":"10.1016/j.nonrwa.2025.104566","DOIUrl":"10.1016/j.nonrwa.2025.104566","url":null,"abstract":"<div><div>This paper concerns subsonic Euler flows in a two-dimensional finitely long slightly curved nozzle under the vertical gravity. Concerning the effect of the vertical gravity, we first establish the existence of subsonic shear flows in the flat nozzle. We then investigate the structural stability of these background subsonic flows under small perturbations of suitable boundary conditions on the entrance and exit and the upper and lower nozzle walls. It can be formulated as a nonlinear boundary value problem for a hyperbolic-elliptic mixed system. The main difficulty is that all the physical quantities are coupled with each other due to the existence of the vertical gravity. The approach is based on the Lagrangian transformation to straighten the streamline and the deformation-curl decomposition to deal with the hyperbolic and elliptic modes in the subsonic region. The key ingredient of the analysis is to solve the associated linearized elliptic boundary value problem with mixed boundary conditions in a weighted Hölder space.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104566"},"PeriodicalIF":1.8,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-11DOI: 10.1016/j.nonrwa.2025.104564
Aiping Zhang, Zesheng Feng, Hongya Gao
This paper deals with a class of non-uniformly polyconvex integral functionals with splitting form in 3-dimensional Euclidean space. Under some structural conditions on the energy density, we prove that each component uα of local minimizer u is locally bounded. Our approach is based on a suitable adaptation of the celebrated De Giorgi’s iterative method, and it relies on an appropriate Caccioppoli-type inequality. Our result can be applied to the polyconvex integralwith suitable functions λ(x) > 0, b(x) ≥ 0 and exponents p, q > 1, r, s ≥ 1.
研究了三维欧几里德空间中一类具有分裂形式的非一致多凸积分泛函。在能量密度的某些结构条件下，证明了局部极小器u的各分量uα是局部有界的。我们的方法是基于著名的De Giorgi的迭代方法的适当改编，它依赖于一个适当的caccioppolii型不等式。我们的结果可以应用于polyconvex积分∫Ω{∑α= 13(λ(x) | Duα| p + | (adj2Du)α| q] + | detDu | r +∑α= 13 b (x) | uα|年代}dxwith合适功能λ(x) 祝辞 0 b (x) ≥ 0和指数p, q 祝辞 1,r, s ≥ 1。
{"title":"Local boundedness for minimizers of some non-uniformly polyconvex integrals","authors":"Aiping Zhang, Zesheng Feng, Hongya Gao","doi":"10.1016/j.nonrwa.2025.104564","DOIUrl":"10.1016/j.nonrwa.2025.104564","url":null,"abstract":"<div><div>This paper deals with a class of non-uniformly polyconvex integral functionals with splitting form in 3-dimensional Euclidean space. Under some structural conditions on the energy density, we prove that each component <em>u<sup>α</sup></em> of local minimizer <em>u</em> is locally bounded. Our approach is based on a suitable adaptation of the celebrated De Giorgi’s iterative method, and it relies on an appropriate Caccioppoli-type inequality. Our result can be applied to the polyconvex integral<span><span><span><math><mrow><msub><mo>∫</mo><mstyle><mi>Ω</mi></mstyle></msub><mrow><mo>{</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><mrow><mo>[</mo><mi>λ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo><mi>D</mi></mrow><msup><mi>u</mi><mi>α</mi></msup><msup><mrow><mo>|</mo></mrow><mi>p</mi></msup><mrow><mo>+</mo><mo>|</mo></mrow><msup><mrow><mo>(</mo><msub><mrow><mrow><mi>a</mi></mrow><mi>d</mi><mi>j</mi></mrow><mn>2</mn></msub><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mi>α</mi></msup><msup><mrow><mo>|</mo></mrow><mi>q</mi></msup><msup><mrow><mo>]</mo><mo>+</mo><mo>|</mo><mi>det</mi><mi>D</mi><mi>u</mi><mo>|</mo></mrow><mi>r</mi></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mo>|</mo><msup><mi>u</mi><mi>α</mi></msup><mo>|</mo></mrow><mi>s</mi></msup><mo>}</mo></mrow><mrow><mi>d</mi></mrow><mi>x</mi></mrow></math></span></span></span>with suitable functions <em>λ</em>(<em>x</em>) > 0, <em>b</em>(<em>x</em>) ≥ 0 and exponents <em>p, q</em> > 1, <em>r, s</em> ≥ 1.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104564"},"PeriodicalIF":1.8,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-10DOI: 10.1016/j.nonrwa.2025.104576
Ken Abe
We consider the stability of magnetohydrostatic (MHS) equilibria in the ideal MHD equations in a bounded and simply connected domain . We show that the set of magnetic energy minimizers with constant helicity (i.e., linear force-free fields) is Lyapunov stable among the weak ideal limits of Leray–Hopf solutions to the viscous and resistive MHD equations.
{"title":"Stability of force-free fields in weak ideal limits of Leray–Hopf solutions I: Linear force-free fields","authors":"Ken Abe","doi":"10.1016/j.nonrwa.2025.104576","DOIUrl":"10.1016/j.nonrwa.2025.104576","url":null,"abstract":"<div><div>We consider the stability of magnetohydrostatic (MHS) equilibria in the ideal MHD equations in a bounded and simply connected domain <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mo>⊂</mo><msup><mi>R</mi><mn>3</mn></msup></mrow></math></span>. We show that the set of magnetic energy minimizers with constant helicity (i.e., linear force-free fields) is Lyapunov stable among the weak ideal limits of Leray–Hopf solutions to the viscous and resistive MHD equations.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104576"},"PeriodicalIF":1.8,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-10DOI: 10.1016/j.nonrwa.2025.104563
Christian Kuehn , Jaeyoung Yoon
Differences in opinion can be seen as distances between individuals, and such differences do not always vanish over time. In this paper, we propose a modeling framework that captures the formation of opinion clusters, based on extensions of the Cucker-Smale and Hegselmann-Krause models to a combined adaptive (or co-evolutionary) network. Reducing our model to a singular limit of fast adaptation, we mathematically analyze the asymptotic behavior of the resulting Laplacian dynamics over various classes of temporal graphs and use these results to explain the behavior of the original proposed adaptive model for fast adaptation. In particular, our approach provides a general methodology for analyzing linear consensus models over time-varying networks that naturally arise as singular limits in many adaptive network models.
{"title":"Adaptive Cucker-Smale networks: Limiting Laplacian time-varying dynamics","authors":"Christian Kuehn , Jaeyoung Yoon","doi":"10.1016/j.nonrwa.2025.104563","DOIUrl":"10.1016/j.nonrwa.2025.104563","url":null,"abstract":"<div><div>Differences in opinion can be seen as distances between individuals, and such differences do not always vanish over time. In this paper, we propose a modeling framework that captures the formation of opinion clusters, based on extensions of the Cucker-Smale and Hegselmann-Krause models to a combined adaptive (or co-evolutionary) network. Reducing our model to a singular limit of fast adaptation, we mathematically analyze the asymptotic behavior of the resulting Laplacian dynamics over various classes of temporal graphs and use these results to explain the behavior of the original proposed adaptive model for fast adaptation. In particular, our approach provides a general methodology for analyzing linear consensus models over time-varying networks that naturally arise as singular limits in many adaptive network models.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104563"},"PeriodicalIF":1.8,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at least one strong time-periodic solution, provided the forcing term is time-periodic. The forcing term does not need to satisfy a smallness condition and is allowed to be arbitrarily large.
{"title":"Time-periodic solutions to an energy balance model coupled with an active fluid under arbitrarily large forces","authors":"Gianmarco Del Sarto , Matthias Hieber , Filippo Palma , Tarek Zöchling","doi":"10.1016/j.nonrwa.2025.104558","DOIUrl":"10.1016/j.nonrwa.2025.104558","url":null,"abstract":"<div><div>This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at least one strong time-periodic solution, provided the forcing term is time-periodic. The forcing term does not need to satisfy a smallness condition and is allowed to be arbitrarily large.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104558"},"PeriodicalIF":1.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-06DOI: 10.1016/j.nonrwa.2025.104565
Dalal Daw , Marko Nedeljkov , Sanja Ružičić
The shallow water system with a single discontinuity of the bottom topography is a simple model of a fluid flow with a sudden change in bottom elevation. The discontinuity is modeled by an increasing step function for simplicity and further application to a specific problem. Using an piecewise continuous approximation with the parameter ε ≪ 1 of the step function at , the Riemann problem for the system is transferred after letting ε → 0 into the minimization of energy problem with the constraints: The approximated solution has to satisfy the Dafermos’s maximal dissipation condition around the discontinuity, while shock and rarefaction waves must have non-positive velocity for x < 0 and non-negative one for x > 0. The application of the procedure is made for the dam-break problem with the additional assumptions that fluid is moving from left to right for x < 0 and the bed is dry for x > 0. The existence of an admissible solution corresponds to the dam failure. A shadow wave is a perturbation that consists of two shock waves with an infinitesimally small area between. It is used to model the fluid behavior near the discontinuity.
{"title":"Maximal energy dissipation principle and the Riemann problem for shallow water flow: Application to the dam-break problem","authors":"Dalal Daw , Marko Nedeljkov , Sanja Ružičić","doi":"10.1016/j.nonrwa.2025.104565","DOIUrl":"10.1016/j.nonrwa.2025.104565","url":null,"abstract":"<div><div>The shallow water system with a single discontinuity of the bottom topography is a simple model of a fluid flow with a sudden change in bottom elevation. The discontinuity is modeled by an increasing step function for simplicity and further application to a specific problem. Using an piecewise continuous approximation with the parameter ε ≪ 1 of the step function at <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>, the Riemann problem for the system is transferred after letting ε → 0 into the minimization of energy problem with the constraints: The approximated solution has to satisfy the Dafermos’s maximal dissipation condition around the discontinuity, while shock and rarefaction waves must have non-positive velocity for <em>x</em> < 0 and non-negative one for <em>x</em> > 0. The application of the procedure is made for the dam-break problem with the additional assumptions that fluid is moving from left to right for <em>x</em> < 0 and the bed is dry for <em>x</em> > 0. The existence of an admissible solution corresponds to the dam failure. A shadow wave is a perturbation that consists of two shock waves with an infinitesimally small area between. It is used to model the fluid behavior near the discontinuity.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104565"},"PeriodicalIF":1.8,"publicationDate":"2025-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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.nonrwa.2025.104559
Yinuo Han , Jianwang Wu , Qian Zhang
<div><div>This paper investigates an attraction-repulsion Navier-Stokes system that incorporates the consumption of chemoattractant and sub-quadratic degradation:<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>n</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>n</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>n</mi><mo>−</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>n</mi><mi>∇</mi><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>n</mi><mi>∇</mi><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>c</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>c</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>c</mi><mo>−</mo><mi>c</mi><mi>n</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>w</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>w</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>n</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>+</mo><mrow><mo>(</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>u</mi><mo>+</mo><mi>∇</mi><mi>P</mi><mo>+</mo><mi>n</mi><mi>∇</mi><mstyle><mi>Λ</mi></mstyle><mo>,</mo><mspace></mspace><mi>∇</mi><mo>·</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>in a bounded and smooth domain <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mo>⊂</mo><msup><mi>R</mi><mn>3</mn></msup></mrow></math></span>, with no-flux/Dirichlet boundary conditions and nonnegative integrable initial data, where <em>f</em> ∈ <em>C</em><sup>1</sup>([0, ∞)) is supposed to generalize standard choices of logistic-type reproduction and degradation, as obtained on letting <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>ρ</mi><mi>s</mi><mo>−</mo><mi>μ</mi><msup><mi>s</mi><mi>α</mi></msup></mrow></math></span> for <em>s</em> ≥ 0, with <em>ρ</em> ≥ 0, <em>μ</em> > 0 and <em>α</em> ∈ (1, 2). In this work, it is demonstrated that, in addition to the fundamental condition that <em>f</em>(0) must be nonnegative, the assumption<span><span><span><math><mtable><mtr><mtd><mrow><mfrac><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>s</mi></mfrac><mo>→</mo><mo>−</mo><mi>∞</mi><mspace></mspace><mspace></mspace><mtext>as</mtext><mspace></mspac
{"title":"Global solvability in a 3D attraction-repulsion Navier-Stokes system with consumption of chemoattractant and sub-quadratic degradation","authors":"Yinuo Han , Jianwang Wu , Qian Zhang","doi":"10.1016/j.nonrwa.2025.104559","DOIUrl":"10.1016/j.nonrwa.2025.104559","url":null,"abstract":"<div><div>This paper investigates an attraction-repulsion Navier-Stokes system that incorporates the consumption of chemoattractant and sub-quadratic degradation:<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>n</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>n</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>n</mi><mo>−</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>n</mi><mi>∇</mi><mi>c</mi><mo>)</mo></mrow><mo>+</mo><mi>∇</mi><mo>·</mo><mrow><mo>(</mo><mi>n</mi><mi>∇</mi><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>c</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>c</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>c</mi><mo>−</mo><mi>c</mi><mi>n</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>w</mi><mi>t</mi></msub><mo>+</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mi>w</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>w</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>n</mi><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>+</mo><mrow><mo>(</mo><mi>u</mi><mo>·</mo><mi>∇</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>u</mi><mo>+</mo><mi>∇</mi><mi>P</mi><mo>+</mo><mi>n</mi><mi>∇</mi><mstyle><mi>Λ</mi></mstyle><mo>,</mo><mspace></mspace><mi>∇</mi><mo>·</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><mi>x</mi><mo>∈</mo><mstyle><mi>Ω</mi></mstyle><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>in a bounded and smooth domain <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mo>⊂</mo><msup><mi>R</mi><mn>3</mn></msup></mrow></math></span>, with no-flux/Dirichlet boundary conditions and nonnegative integrable initial data, where <em>f</em> ∈ <em>C</em><sup>1</sup>([0, ∞)) is supposed to generalize standard choices of logistic-type reproduction and degradation, as obtained on letting <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>ρ</mi><mi>s</mi><mo>−</mo><mi>μ</mi><msup><mi>s</mi><mi>α</mi></msup></mrow></math></span> for <em>s</em> ≥ 0, with <em>ρ</em> ≥ 0, <em>μ</em> > 0 and <em>α</em> ∈ (1, 2). In this work, it is demonstrated that, in addition to the fundamental condition that <em>f</em>(0) must be nonnegative, the assumption<span><span><span><math><mtable><mtr><mtd><mrow><mfrac><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>s</mi></mfrac><mo>→</mo><mo>−</mo><mi>∞</mi><mspace></mspace><mspace></mspace><mtext>as</mtext><mspace></mspac","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"90 ","pages":"Article 104559"},"PeriodicalIF":1.8,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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