Pub Date : 2025-09-17DOI: 10.1016/j.nonrwa.2025.104506
Laura M. Fernández–Pardo, Jorge Rodríguez–López
We present a version of Krasnosel’skiĭ fixed point theorem for operators acting on Cartesian products of normed linear spaces, under cone-compression and cone-expansion conditions of norm type. Our approach, based on the fixed point index theory in cones, guarantees the existence of a coexistence fixed point, that is, one with nontrivial components. As an application, we prove the existence of periodic solutions with strictly positive components for a system of second-order differential equations. In particular, we address cases involving singular nonlinearities and hybrid terms, characterized by sublinear behavior in one component and superlinear behavior in the other.
{"title":"Component-wise Krasnosel’skii type fixed point theorem in product spaces and applications","authors":"Laura M. Fernández–Pardo, Jorge Rodríguez–López","doi":"10.1016/j.nonrwa.2025.104506","DOIUrl":"10.1016/j.nonrwa.2025.104506","url":null,"abstract":"<div><div>We present a version of Krasnosel’skiĭ fixed point theorem for operators acting on Cartesian products of normed linear spaces, under cone-compression and cone-expansion conditions of norm type. Our approach, based on the fixed point index theory in cones, guarantees the existence of a coexistence fixed point, that is, one with nontrivial components. As an application, we prove the existence of periodic solutions with strictly positive components for a system of second-order differential equations. In particular, we address cases involving singular nonlinearities and hybrid terms, characterized by sublinear behavior in one component and superlinear behavior in the other.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104506"},"PeriodicalIF":1.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099837","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 : 2025-09-17DOI: 10.1016/j.nonrwa.2025.104504
Yuri Soga
We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing:where is a ball in two-dimensional space and with nonnegative radially symmetric initial data. In the case of the Keller–Segel system which has similar mathematical structures to our system, it was shown that solutions blow up in finite time if and only if for the first component diverges in finite time. On the other hand, focusing on the variational structure induced by the signal-dependent motility function , we show that the unboundedness of for the second component gives rise to blowup solutions in infinite time under the assumption of radial symmetry. Moreover we prove mass concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system develop a singularity like a Dirac delta function in infinite time. Here we investigate the weight of this singularity. Consequently it is shown that mass quantization may not occur; that is, the weight of the singularity can exceed under the assumption of a uniform-in-time lower bound for a Lyapunov functional. This type of behavior cannot be observed in the Keller–Segel system.
{"title":"A sufficient condition for absence of mass quantization in a chemotaxis system with local sensing","authors":"Yuri Soga","doi":"10.1016/j.nonrwa.2025.104504","DOIUrl":"10.1016/j.nonrwa.2025.104504","url":null,"abstract":"<div><div>We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing:<span><span><span><math><mtable><mtr><mtd><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mrow><mo>(</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>v</mi></mrow></msup><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mi>in</mi></mrow><mspace></mspace><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>v</mi><mi>t</mi></msub><mo>=</mo><mstyle><mi>Δ</mi></mstyle><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mspace></mspace><mrow><mi>in</mi></mrow><mspace></mspace><mstyle><mi>Ω</mi></mstyle><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></math></span></span></span>where <span><math><mstyle><mi>Ω</mi></mstyle></math></span> is a ball in two-dimensional space and with nonnegative radially symmetric initial data. In the case of the Keller–Segel system which has similar mathematical structures to our system, it was shown that solutions blow up in finite time if and only if <span><math><mrow><mi>L</mi><mi>log</mi><mi>L</mi></mrow></math></span> for the first component <span><math><mi>u</mi></math></span> diverges in finite time. On the other hand, focusing on the variational structure induced by the signal-dependent motility function <span><math><msup><mi>e</mi><mrow><mo>−</mo><mi>v</mi></mrow></msup></math></span>, we show that the unboundedness of <span><math><mrow><msub><mo>∫</mo><mstyle><mi>Ω</mi></mstyle></msub><msup><mi>e</mi><mi>v</mi></msup><mi>d</mi><mi>x</mi></mrow></math></span> for the second component <span><math><mi>v</mi></math></span> gives rise to blowup solutions in infinite time under the assumption of radial symmetry. Moreover we prove mass concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system develop a singularity like a Dirac delta function in infinite time. Here we investigate the weight of this singularity. Consequently it is shown that mass quantization may not occur; that is, the weight of the singularity can exceed <span><math><mrow><mn>8</mn><mi>π</mi></mrow></math></span> under the assumption of a uniform-in-time lower bound for a Lyapunov functional. This type of behavior cannot be observed in the Keller–Segel system.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104504"},"PeriodicalIF":1.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099836","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 : 2025-09-16DOI: 10.1016/j.nonrwa.2025.104502
Yaobin Tang, Zhenzhen Li, Binxiang Dai
The paper considers the dynamical behaviors of two competing species for the case of weak competition with nonlocal dispersal and seasonal succession. We first derive the existence and non-existence of traveling waves connecting the trivial equilibrium and the positive periodic solution by using the method of upper-lower solutions and the asymptotic fixed point theorem. Then we obtain the asymptotic spreading properties of the two competing species with compactly supported initial conditions. Our results demonstrate that a competitively weaker species with a faster spreading speed can drive a competitively stronger but slower-spreading species to extinction.
{"title":"Propagation of nonlocal dispersal competition model with seasonal succession","authors":"Yaobin Tang, Zhenzhen Li, Binxiang Dai","doi":"10.1016/j.nonrwa.2025.104502","DOIUrl":"10.1016/j.nonrwa.2025.104502","url":null,"abstract":"<div><div>The paper considers the dynamical behaviors of two competing species for the case of weak competition with nonlocal dispersal and seasonal succession. We first derive the existence and non-existence of traveling waves connecting the trivial equilibrium and the positive periodic solution by using the method of upper-lower solutions and the asymptotic fixed point theorem. Then we obtain the asymptotic spreading properties of the two competing species with compactly supported initial conditions. Our results demonstrate that a competitively weaker species with a faster spreading speed can drive a competitively stronger but slower-spreading species to extinction.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104502"},"PeriodicalIF":1.8,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099450","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 : 2025-09-12DOI: 10.1016/j.nonrwa.2025.104460
Shi-Ke Hu , Jiawei Huo , Rong Yuan , Hai-Feng Huo
This article studies a nonlocal dispersal two-strain epidemic model with Neumann boundary condition in a heterogeneous environment. We define the basic reproduction number for this model and demonstrate that the disease will persist if the basic reproduction number is bigger than one, but will diminish otherwise. We also investigate the competitive exclusion of two strains in various local distributions of transmission and recovery rates. Furthermore, we study the boundedness and existence of the coexist endemic steady-state solution of this model with respect to the nonlocal dispersal rates. Particularly, it is found that, in comparison to the models with random diffusion, two strains with nonlocal dispersal are more liable to coexist, and the disease are more difficult to be controlled via reducing the movement of individuals.
{"title":"Spatial dynamics of a two-strain epidemic model with nonlocal dispersal","authors":"Shi-Ke Hu , Jiawei Huo , Rong Yuan , Hai-Feng Huo","doi":"10.1016/j.nonrwa.2025.104460","DOIUrl":"10.1016/j.nonrwa.2025.104460","url":null,"abstract":"<div><div>This article studies a nonlocal dispersal two-strain epidemic model with Neumann boundary condition in a heterogeneous environment. We define the basic reproduction number for this model and demonstrate that the disease will persist if the basic reproduction number is bigger than one, but will diminish otherwise. We also investigate the competitive exclusion of two strains in various local distributions of transmission and recovery rates. Furthermore, we study the boundedness and existence of the coexist endemic steady-state solution of this model with respect to the nonlocal dispersal rates. Particularly, it is found that, in comparison to the models with random diffusion, two strains with nonlocal dispersal are more liable to coexist, and the disease are more difficult to be controlled via reducing the movement of individuals.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104460"},"PeriodicalIF":1.8,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048421","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 : 2025-09-09DOI: 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":"2025-09-09","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}
Pub 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":"2025-09-03","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 : 2025-09-02DOI: 10.1016/j.nonrwa.2025.104482
Yassin El Karrouchi, Tobias Weth
Inspired by recent work of Ferone and Volzone (2021), we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti’s comparison principle in the context of Dirichlet–Poisson problems for the fractional Laplacian in the unit ball , . In particular, our results imply a universal failure of the classical pointwise Talenti inequality in the fractional radial context. In contrast, a boundary Talenti type inequality holds for radial functions in the higher order case .
{"title":"On a fractional boundary version of Talenti’s inequality in the unit ball","authors":"Yassin El Karrouchi, Tobias Weth","doi":"10.1016/j.nonrwa.2025.104482","DOIUrl":"10.1016/j.nonrwa.2025.104482","url":null,"abstract":"<div><div>Inspired by recent work of Ferone and Volzone (2021), we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti’s comparison principle in the context of Dirichlet–Poisson problems for the fractional Laplacian <span><math><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup></math></span> in the unit ball <span><math><mrow><mi>Ω</mi><mo>=</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>s</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. In particular, our results imply a universal failure of the classical pointwise Talenti inequality in the fractional radial context. In contrast, a boundary Talenti type inequality holds for radial functions in the higher order case <span><math><mrow><mi>s</mi><mo>></mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104482"},"PeriodicalIF":1.8,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932349","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 : 2025-08-31DOI: 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":"2025-08-31","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 : 2025-08-19DOI: 10.1016/j.nonrwa.2025.104479
Felipe Breton , Carlos Martínez
Phosphorus release from sediments in lakes can trigger harmful algal blooms, significantly impacting lake ecosystems and necessitating effective management strategies. This study presents a mathematical analysis of a dynamic model incorporating such internal phosphorus loadings and their impact on algal growth. The model accounts for both light and phosphorus limitations on cell growth, as well as the seasonal variability of temperature and light, leading to a periodically forced non-linear system of ordinary differential equations. Using the theory of periodic semiflows and Floquet multipliers, we establish both necessary and sufficient conditions for long-term survival of algae (uniform persistence). This approach provides a threshold result for algae survival and the existence of a non-trivial periodic solution. Through numerical simulations, we illustrate our results and provide insights into the role of internal phosphorus loadings. In particular, our simulations illustrate that an integrated strategy reducing both watershed inflows and sediment phosphorus outperforms measures that focus on just one source of phosphorus.
{"title":"Persistence of harmful algal blooms under conditions of internal phosphorus loading","authors":"Felipe Breton , Carlos Martínez","doi":"10.1016/j.nonrwa.2025.104479","DOIUrl":"10.1016/j.nonrwa.2025.104479","url":null,"abstract":"<div><div>Phosphorus release from sediments in lakes can trigger harmful algal blooms, significantly impacting lake ecosystems and necessitating effective management strategies. This study presents a mathematical analysis of a dynamic model incorporating such internal phosphorus loadings and their impact on algal growth. The model accounts for both light and phosphorus limitations on cell growth, as well as the seasonal variability of temperature and light, leading to a periodically forced non-linear system of ordinary differential equations. Using the theory of periodic semiflows and Floquet multipliers, we establish both necessary and sufficient conditions for long-term survival of algae (uniform persistence). This approach provides a threshold result for algae survival and the existence of a non-trivial periodic solution. Through numerical simulations, we illustrate our results and provide insights into the role of internal phosphorus loadings. In particular, our simulations illustrate that an integrated strategy reducing both watershed inflows and sediment phosphorus outperforms measures that focus on just one source of phosphorus.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104479"},"PeriodicalIF":1.8,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865885","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 : 2025-08-16DOI: 10.1016/j.nonrwa.2025.104478
Changchun Liu, Yue Zhou
In this paper, we consider a angiogenesis system with -Laplacian diffusion in a bounded domain with smooth boundary. For all we prove the existence of global strong solution.
{"title":"Global solutions to the angiogenesis system with p-Laplacian diffusion","authors":"Changchun Liu, Yue Zhou","doi":"10.1016/j.nonrwa.2025.104478","DOIUrl":"10.1016/j.nonrwa.2025.104478","url":null,"abstract":"<div><div>In this paper, we consider a angiogenesis system with <span><math><mi>p</mi></math></span>-Laplacian diffusion <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mfenced><mrow><mrow><mo>(</mo><msup><mrow><mfenced><mrow><mo>∇</mo><mi>u</mi></mrow></mfenced></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi></mrow></mfenced><mo>−</mo><mi>χ</mi><mo>∇</mo><mi>⋅</mi><mfenced><mrow><mi>u</mi><mo>∇</mo><mi>v</mi></mrow></mfenced><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace></mtd><mtd><mspace></mspace><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mrow><mo>(</mo><mi>N</mi><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> with smooth boundary. For all <span><span><span><math><mrow><mi>p</mi><mo>></mo><msub><mrow><mi>p</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>=</mo><mo>max</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mn>4</mn><mi>N</mi></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>}</mo></mrow><mo>,</mo></mrow></math></span></span></span>we prove the existence of global strong solution.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104478"},"PeriodicalIF":1.8,"publicationDate":"2025-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144858431","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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