Pub Date : 2026-04-01Epub Date: 2025-07-25DOI: 10.1016/j.nonrwa.2025.104459
Wenwen Fu, Qingqing Liu
In this paper, we are concerned with a hyperbolic–parabolic–elliptic vasculogenesis model in the half-space under outflow boundary conditions. It is shown that the planar stationary solution is stable with respect to small perturbations in and the perturbations decay in norm as , provided that the magnitude of the stationary solution is sufficiently small. This result is proved by basic energy estimates. Compared with Navier–Stokes equations, we have effectively dealt with the coupling between the fluid quantities and chemoattractant in the vasculogenesis model.
{"title":"Stability of planar stationary solution for outflow problem on the viscous vasculogenesis model","authors":"Wenwen Fu, Qingqing Liu","doi":"10.1016/j.nonrwa.2025.104459","DOIUrl":"10.1016/j.nonrwa.2025.104459","url":null,"abstract":"<div><div>In this paper, we are concerned with a hyperbolic–parabolic–elliptic vasculogenesis model in the half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> under outflow boundary conditions. It is shown that the planar stationary solution is stable with respect to small perturbations in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the perturbations decay in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> norm as <span><math><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, provided that the magnitude of the stationary solution is sufficiently small. This result is proved by basic energy estimates. Compared with Navier–Stokes equations, we have effectively dealt with the coupling between the fluid quantities and chemoattractant in the vasculogenesis model.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104459"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703763","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 : 2026-04-01Epub Date: 2025-07-26DOI: 10.1016/j.nonrwa.2025.104467
Håkan Andréasson , Markus Kunze , Gerhard Rein
We give a new proof for the existence of spherically symmetric steady states to the Vlasov-Poisson system, following a strategy that has been used successfully to approximate axially symmetric solutions numerically, both to the Vlasov–Poisson system and to the Einstein–Vlasov system. There are several reasons why a mathematical analysis of this numerical scheme is important. A generalization of the present result to the case of flat axially symmetric solutions would prove that the steady states obtained numerically in Andréasson and Rein (2015) do exist. Moreover, in the relativistic case the question whether a steady state can be obtained by this scheme seems to be related to its dynamical stability. This motivates the desire for a deeper understanding of this strategy.
{"title":"Steady states of the spherically symmetric Vlasov-Poisson system as fixed points of a mass-preserving algorithm","authors":"Håkan Andréasson , Markus Kunze , Gerhard Rein","doi":"10.1016/j.nonrwa.2025.104467","DOIUrl":"10.1016/j.nonrwa.2025.104467","url":null,"abstract":"<div><div>We give a new proof for the existence of spherically symmetric steady states to the Vlasov-Poisson system, following a strategy that has been used successfully to approximate axially symmetric solutions numerically, both to the Vlasov–Poisson system and to the Einstein–Vlasov system. There are several reasons why a mathematical analysis of this numerical scheme is important. A generalization of the present result to the case of flat axially symmetric solutions would prove that the steady states obtained numerically in Andréasson and Rein (2015) do exist. Moreover, in the relativistic case the question whether a steady state can be obtained by this scheme seems to be related to its dynamical stability. This motivates the desire for a deeper understanding of this strategy.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104467"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144713222","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 : 2026-04-01Epub Date: 2025-09-17DOI: 10.1016/j.nonrwa.2025.104505
Julia Calatayud , Marc Jornet , Carla M.A. Pinto
We propose a new mathematical model to capture the overlapping dynamics of dengue and COVID-19 infections in a susceptible population, based on a nonlinear system of ordinary differential equations. First, we calculate the basic reproduction number and present its use in the analysis of outbreaks, long-term dynamics, and parameter sensitivity. Then, we introduce an Itô stochastic version of the system and conduct numerical simulations to explore its behavior, which generalizes the deterministic counterpart. The model is validated with real-world data from Colombia, employing different approaches: global and sub-stages fitting. We describe the emerging challenges, namely, unidentifiable parameters and limited data availability. To simplify the least-squares optimization process, certain parameters were previously fixed. Consequently, the model’s results should be interpreted with caution. Overcoming these limitations will be critical to advance epidemic modeling.
{"title":"Patterns of dengue and SARS-CoV-2 coinfection in the light of deterministic and stochastic models","authors":"Julia Calatayud , Marc Jornet , Carla M.A. Pinto","doi":"10.1016/j.nonrwa.2025.104505","DOIUrl":"10.1016/j.nonrwa.2025.104505","url":null,"abstract":"<div><div>We propose a new mathematical model to capture the overlapping dynamics of dengue and COVID-19 infections in a susceptible population, based on a nonlinear system of ordinary differential equations. First, we calculate the basic reproduction number and present its use in the analysis of outbreaks, long-term dynamics, and parameter sensitivity. Then, we introduce an Itô stochastic version of the system and conduct numerical simulations to explore its behavior, which generalizes the deterministic counterpart. The model is validated with real-world data from Colombia, employing different approaches: global and sub-stages fitting. We describe the emerging challenges, namely, unidentifiable parameters and limited data availability. To simplify the least-squares optimization process, certain parameters were previously fixed. Consequently, the model’s results should be interpreted with caution. Overcoming these limitations will be critical to advance epidemic modeling.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104505"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099834","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 : 2026-04-01Epub Date: 2025-08-31DOI: 10.1016/j.nonrwa.2025.104490
Huashui Zhan
By the Crocco transformation, the boundary layer system of the viscous incompressible flow is transferred to a strong degenerate parabolic equation with a nonlinear boundary value condition, referred as the Prandtl boundary layer equation. The key technique in this paper involves applying the reciprocal transformation to convert the Prandtl boundary layer equation into a degenerate parabolic equation in divergent form. The main challenge arises on account of that the reciprocal transformation renders the initial value condition unbounded. To address this, a new unknown function is introduced, and the partial differential equation for is derived. For this new equation, the existence of these BV entropy solutions are proved by the parabolically regularized method, the maximal value principle is used to obtain the -estimate. Under certain restrictions on the data of the Prandtl system, the stability of entropy solutions is demonstrated using different boundary value conditions. Consequently, under the Oleǐnik assumption and the monotonicity condition, the two-dimensional Prandtl boundary layer system is shown to be well-posed through the inverse Crocco transformation.
{"title":"Stability of the Prandtl boundary layer equation under various boundary conditions","authors":"Huashui Zhan","doi":"10.1016/j.nonrwa.2025.104490","DOIUrl":"10.1016/j.nonrwa.2025.104490","url":null,"abstract":"<div><div>By the Crocco transformation, the boundary layer system of the viscous incompressible flow is transferred to a strong degenerate parabolic equation with a nonlinear boundary value condition, referred as the Prandtl boundary layer equation. The key technique in this paper involves applying the reciprocal transformation to convert the Prandtl boundary layer equation into a degenerate parabolic equation in divergent form. The main challenge arises on account of that the reciprocal transformation renders the initial value condition unbounded. To address this, a new unknown function <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is introduced, and the partial differential equation for <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is derived. For this new equation, the existence of these BV entropy solutions are proved by the parabolically regularized method, the maximal value principle is used to obtain the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-estimate. Under certain restrictions on the data of the Prandtl system, the stability of entropy solutions is demonstrated using different boundary value conditions. Consequently, under the Oleǐnik assumption and the monotonicity condition, the two-dimensional Prandtl boundary layer system is shown to be well-posed through the inverse Crocco transformation.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104490"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144921984","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 : 2026-04-01Epub Date: 2025-09-03DOI: 10.1016/j.nonrwa.2025.104491
Martha Alvarez-Ramírez , Johanna D. García-Saldaña , Jaume Llibre
In mechanics jerk is the rate of change of an object’s acceleration over time. Thus a jerk equation is a differential equation of the form , where , , and represent the position, velocity, acceleration, and jerk, respectively. The jerk differential equation can be written as the jerk differential system in . In this paper we study the jerk differential system with , previously studied by other authors showing that this system can exhibit chaos for some values of its parameters. When the parameters the -axis is filled with zero-Hopf equilibria, and all the other orbits are periodic. Here we prove analytically the existence of two families of periodic orbits for sufficiently small values of the parameters and . One family bifurcates from the non-isolated zero-Hopf equilibrium of the jerk system with , while the other family bifurcates from a periodic orbit of the jerk system with .
{"title":"Integrability and periodic orbits of a 3D jerk system with two quadratic nonlinearities","authors":"Martha Alvarez-Ramírez , Johanna D. García-Saldaña , Jaume Llibre","doi":"10.1016/j.nonrwa.2025.104491","DOIUrl":"10.1016/j.nonrwa.2025.104491","url":null,"abstract":"<div><div>In mechanics jerk is the rate of change of an object’s acceleration over time. Thus a jerk equation is a differential equation of the form <span><math><mrow><mover><mrow><mi>x</mi></mrow><mrow><mo>⃛</mo></mrow></mover><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>̇</mo></mrow></mover><mo>,</mo><mover><mrow><mi>x</mi></mrow><mrow><mo>̈</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>x</mi></math></span>, <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>̇</mo></mrow></mover></math></span>, <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>̈</mo></mrow></mover></math></span> and <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>⃛</mo></mrow></mover></math></span> represent the position, velocity, acceleration, and jerk, respectively. The jerk differential equation can be written as the jerk differential system <span><math><mrow><mover><mrow><mi>x</mi></mrow><mrow><mo>̇</mo></mrow></mover><mo>=</mo><mi>y</mi><mo>,</mo><mspace></mspace><mover><mrow><mi>y</mi></mrow><mrow><mo>̇</mo></mrow></mover><mo>=</mo><mi>z</mi><mo>,</mo><mspace></mspace><mover><mrow><mi>z</mi></mrow><mrow><mo>̇</mo></mrow></mover><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. In this paper we study the jerk differential system with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>a</mi><mi>x</mi><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>y</mi><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, previously studied by other authors showing that this system can exhibit chaos for some values of its parameters. When the parameters <span><math><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span> the <span><math><mi>x</mi></math></span>-axis is filled with zero-Hopf equilibria, and all the other orbits are periodic. Here we prove analytically the existence of two families of periodic orbits for sufficiently small values of the parameters <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span>. One family bifurcates from the non-isolated zero-Hopf equilibrium <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></math></span> of the jerk system with <span><math><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span>, while the other family bifurcates from a periodic orbit of the jerk system with <span><math><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104491"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932266","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 : 2026-04-01Epub Date: 2025-07-29DOI: 10.1016/j.nonrwa.2025.104461
Tomoyuki Nakatsuka
This paper is devoted to the study of the time-periodic problem for the Oberbeck–Boussinesq system in the whole space. Our investigation is based on the reformulation of the time-periodic problem and does not depend on the analysis of the initial value problem. We construct a time-periodic solution with more information on its structure than the solutions in preceding studies. We also prove that our solution, small in an appropriate sense, is unique in the class of solutions having slightly more regularity.
{"title":"Existence and uniqueness of time-periodic solutions to the Oberbeck–Boussinesq system","authors":"Tomoyuki Nakatsuka","doi":"10.1016/j.nonrwa.2025.104461","DOIUrl":"10.1016/j.nonrwa.2025.104461","url":null,"abstract":"<div><div>This paper is devoted to the study of the time-periodic problem for the Oberbeck–Boussinesq system in the whole space. Our investigation is based on the reformulation of the time-periodic problem and does not depend on the analysis of the initial value problem. We construct a time-periodic solution with more information on its structure than the solutions in preceding studies. We also prove that our solution, small in an appropriate sense, is unique in the class of solutions having slightly more regularity.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104461"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722113","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 : 2026-04-01Epub Date: 2025-08-05DOI: 10.1016/j.nonrwa.2025.104471
Wan-Tong Li, Juan Qiu, Ming-Zhen Xin, Xu-Dong Zhao
This paper is concerned with the invasion waves for a class of multi-species non-cooperative systems with nonlocal dispersal. We first establish a sharp existence result of the weak traveling wave solution connected the semi-trivial equilibrium for a general multi-species nonlocal dispersal system by Schauder’s fixed-point theorem. And then we apply this result to discuss the traveling wave solutions for a disease-transmission model and a predator–prey model respectively, where we prove that the weak traveling wave solutions connect the positive equilibrium with the help of Lyapunov functional. To get the asymptotic behavior of traveling wave solutions at , we have to overcome the difficulties brought by the nonlocal dispersal and the non-cooperative of systems themselves.
{"title":"Invasion waves for a class of multi-species non-cooperative systems with nonlocal dispersal","authors":"Wan-Tong Li, Juan Qiu, Ming-Zhen Xin, Xu-Dong Zhao","doi":"10.1016/j.nonrwa.2025.104471","DOIUrl":"10.1016/j.nonrwa.2025.104471","url":null,"abstract":"<div><div>This paper is concerned with the invasion waves for a class of multi-species non-cooperative systems with nonlocal dispersal. We first establish a sharp existence result of the weak traveling wave solution connected the semi-trivial equilibrium for a general multi-species nonlocal dispersal system by Schauder’s fixed-point theorem. And then we apply this result to discuss the traveling wave solutions for a disease-transmission model and a predator–prey model respectively, where we prove that the weak traveling wave solutions connect the positive equilibrium with the help of Lyapunov functional. To get the asymptotic behavior of traveling wave solutions at <span><math><mrow><mo>+</mo><mi>∞</mi></mrow></math></span>, we have to overcome the difficulties brought by the nonlocal dispersal and the non-cooperative of systems themselves.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104471"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772650","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 : 2026-04-01Epub Date: 2025-09-09DOI: 10.1016/j.nonrwa.2025.104489
Sophia Hertrich , Tao Huang , Diego Yépez , Kun Zhao
In this research, we study the existence of stationary solutions with vacuum to a hyperbolic–parabolic chemotaxis model with nonlinear pressure in dimension two that describes vasculogenesis. We seek radially symmetric solutions in the whole space, in which the system will be reduced to a system of ODE’s on . The fundamental solutions to the ODE system are the Bessel functions of different types. We find two nontrivial solutions. One is formed by half bump (positive density region) starting at and a region of vacuum on the right. Another one is a full nonsymmetric bump away from . These solutions bear certain resemblance to in vitro vascular network and the numerically produced structure by Gamba et al. (2003). We also show the nonexistence of full bump starting at and nonexistence of full symmetric bump away from .
{"title":"Stationary solutions with vacuum for a hyperbolic–parabolic chemotaxis model in dimension two","authors":"Sophia Hertrich , Tao Huang , Diego Yépez , Kun Zhao","doi":"10.1016/j.nonrwa.2025.104489","DOIUrl":"10.1016/j.nonrwa.2025.104489","url":null,"abstract":"<div><div>In this research, we study the existence of stationary solutions with vacuum to a hyperbolic–parabolic chemotaxis model with nonlinear pressure in dimension two that describes vasculogenesis. We seek radially symmetric solutions in the whole space, in which the system will be reduced to a system of ODE’s on <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>. The fundamental solutions to the ODE system are the Bessel functions of different types. We find two nontrivial solutions. One is formed by half bump (positive density region) starting at <span><math><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span> and a region of vacuum on the right. Another one is a full nonsymmetric bump away from <span><math><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span>. These solutions bear certain resemblance to <em>in vitro</em> vascular network and the numerically produced structure by Gamba et al. (2003). We also show the nonexistence of full bump starting at <span><math><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span> and nonexistence of full symmetric bump away from <span><math><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104489"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019231","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 paper is devoted to the analysis of the global stability of the chemostat system with a perturbation term representing a general form of exchange between species. This conversion term depends not only on species and substrate concentrations, but also on a positive perturbation parameter. After expressing the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there exists a positive threshold for the perturbation parameter below which the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman concerning perturbations of a globally stable steady-state. Properties of the steady-states and numerical simulations of the system’s asymptotic behavior complement this study for two types of perturbation terms between the species.
{"title":"Global stability of perturbed chemostat systems","authors":"Claudia Alvarez-Latuz , Térence Bayen , Jérôme Coville","doi":"10.1016/j.nonrwa.2025.104509","DOIUrl":"10.1016/j.nonrwa.2025.104509","url":null,"abstract":"<div><div>This paper is devoted to the analysis of the global stability of the chemostat system with a perturbation term representing a general form of exchange between species. This conversion term depends not only on species and substrate concentrations, but also on a positive perturbation parameter. After expressing the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there exists a positive threshold for the perturbation parameter below which the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman concerning perturbations of a globally stable steady-state. Properties of the steady-states and numerical simulations of the system’s asymptotic behavior complement this study for two types of perturbation terms between the species.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104509"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117683","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 : 2026-04-01Epub Date: 2025-08-06DOI: 10.1016/j.nonrwa.2025.104470
Qingqing Peng , Yikan Liu
This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in () with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the system.
{"title":"Exponential stability for an infinite memory wave equation with frictional damping and logarithmic nonlinear terms","authors":"Qingqing Peng , Yikan Liu","doi":"10.1016/j.nonrwa.2025.104470","DOIUrl":"10.1016/j.nonrwa.2025.104470","url":null,"abstract":"<div><div>This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> (<span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the system.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104470"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780442","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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