Pub Date : 2025-09-26DOI: 10.1016/j.nonrwa.2025.104511
Qiao Liu, Zhongbao Zuo
<div><div>We study partial regularity and the upper Minkowski dimension of potential singularities for suitable weak solutions to the 3d co-rotational Beris-Edwards system for the nematic liquid crystal flows with Landau-de Gennes potential. Precisely, we establish that there exists a <span><math><mrow><mrow><mi>ε</mi></mrow><mo>></mo><mn>0</mn></mrow></math></span> such that if <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mrow><mi>P</mi></mrow><mo>)</mo></mrow></math></span> is a suitable weak solution, and satisfies<span><span><span><math><mrow><msup><mi>r</mi><mrow><mo>−</mo><mfrac><mrow><mn>6</mn><mi>α</mi></mrow><mrow><mn>7</mn><mi>α</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></msup><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mn>0</mn></msub><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><msub><mi>t</mi><mn>0</mn></msub></msubsup><msup><mrow><mo>(</mo><mo>∥</mo><mo>(</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mn>2</mn></msup><msup><mrow><mo>,</mo><mo>|</mo><mi>∇</mi><mi>Q</mi><mo>|</mo></mrow><mn>2</mn></msup><msubsup><mrow><mo>)</mo><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>α</mi></msup><mrow><mo>(</mo><msub><mi>B</mi><mi>r</mi></msub><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mi>β</mi></msubsup><mo>+</mo><msubsup><mrow><mo>∥</mo><mrow><mi>P</mi></mrow><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>α</mi></msup><mrow><mo>(</mo><msub><mi>B</mi><mi>r</mi></msub><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mi>β</mi></msubsup><mo>)</mo><mrow><mi>d</mi></mrow><mi>t</mi><mo>≤</mo><mrow><mi>ε</mi></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mfrac><mn>6</mn><mn>5</mn></mfrac><mo>,</mo><mn>2</mn><mo>]</mo></mrow></math></span> and <span><math><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>α</mi></mrow><mrow><mn>7</mn><mi>α</mi><mo>−</mo><mn>6</mn></mrow></mfrac><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></math></span>, then <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow></math></span> is regular at <span><math><msub><mi>z</mi><mn>0</mn></msub></math></span>. Based upon the regularity result above, we then prove the upper Minkowski dimension of the potential singularities for any suitable weak solution is at most <span><math><mrow><mfrac><mn>975</mn><mn>758</mn></mfrac><mrow><mo>(</mo><mo>≈</mo><mn>1.286</mn><mo>)</mo></mrow></mrow></math></span>. Additionally, if <span><math><mrow><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>∇</mi><mi>Q</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mi>L</mi><mi>q</mi></msup><mrow><mo>(</mo><msup><mi>R</mi><mn>3</mn></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mfrac><mn>2</mn><mi>p</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mi>q</mi></
{"title":"Partial regularity and the upper Minkowski dimension of singularities for suitable weak solutions to the 3D co-rotational Beris-Edwards system","authors":"Qiao Liu, Zhongbao Zuo","doi":"10.1016/j.nonrwa.2025.104511","DOIUrl":"10.1016/j.nonrwa.2025.104511","url":null,"abstract":"<div><div>We study partial regularity and the upper Minkowski dimension of potential singularities for suitable weak solutions to the 3d co-rotational Beris-Edwards system for the nematic liquid crystal flows with Landau-de Gennes potential. Precisely, we establish that there exists a <span><math><mrow><mrow><mi>ε</mi></mrow><mo>></mo><mn>0</mn></mrow></math></span> such that if <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mrow><mi>P</mi></mrow><mo>)</mo></mrow></math></span> is a suitable weak solution, and satisfies<span><span><span><math><mrow><msup><mi>r</mi><mrow><mo>−</mo><mfrac><mrow><mn>6</mn><mi>α</mi></mrow><mrow><mn>7</mn><mi>α</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></msup><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mn>0</mn></msub><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><msub><mi>t</mi><mn>0</mn></msub></msubsup><msup><mrow><mo>(</mo><mo>∥</mo><mo>(</mo><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mn>2</mn></msup><msup><mrow><mo>,</mo><mo>|</mo><mi>∇</mi><mi>Q</mi><mo>|</mo></mrow><mn>2</mn></msup><msubsup><mrow><mo>)</mo><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>α</mi></msup><mrow><mo>(</mo><msub><mi>B</mi><mi>r</mi></msub><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mi>β</mi></msubsup><mo>+</mo><msubsup><mrow><mo>∥</mo><mrow><mi>P</mi></mrow><mo>∥</mo></mrow><mrow><msup><mi>L</mi><mi>α</mi></msup><mrow><mo>(</mo><msub><mi>B</mi><mi>r</mi></msub><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><mi>β</mi></msubsup><mo>)</mo><mrow><mi>d</mi></mrow><mi>t</mi><mo>≤</mo><mrow><mi>ε</mi></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>α</mi><mo>∈</mo><mo>[</mo><mfrac><mn>6</mn><mn>5</mn></mfrac><mo>,</mo><mn>2</mn><mo>]</mo></mrow></math></span> and <span><math><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>α</mi></mrow><mrow><mn>7</mn><mi>α</mi><mo>−</mo><mn>6</mn></mrow></mfrac><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></math></span>, then <span><math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow></math></span> is regular at <span><math><msub><mi>z</mi><mn>0</mn></msub></math></span>. Based upon the regularity result above, we then prove the upper Minkowski dimension of the potential singularities for any suitable weak solution is at most <span><math><mrow><mfrac><mn>975</mn><mn>758</mn></mfrac><mrow><mo>(</mo><mo>≈</mo><mn>1.286</mn><mo>)</mo></mrow></mrow></math></span>. Additionally, if <span><math><mrow><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>∇</mi><mi>Q</mi><mo>)</mo></mrow><mo>∈</mo><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mi>L</mi><mi>q</mi></msup><mrow><mo>(</mo><msup><mi>R</mi><mn>3</mn></msup><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mfrac><mn>2</mn><mi>p</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mi>q</mi></","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104511"},"PeriodicalIF":1.8,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157518","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}
<div><div>In this work we study a class of discontinuous hybrid piecewise differential systems formed by two Hamiltonian systems that we named piecewise hybrid Hamiltonian systems. More precisely, we consider the differential systems with Hamiltonian functions <span><span><span><span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>y</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>5</mn></mrow></msub><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>A</mi><mo>,</mo><mtext>if</mtext><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span></span><span><span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>y</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>5</mn></mrow></msub><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>B</mi><mo>,</mo><mtext>if</mtext><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math></span></span></span></span>with reset maps <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mi>x</mi></mrow></math></span> on <span><math><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>r</mi><mi>y</mi></mrow></math></span> on <span><math><mrow><mi>y</mi><mo>≥</mo><mn>0</mn></mrow></math></span> for <span><math><mrow><mn>0</mn><mo><</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo><</mo><mn>1</mn></mrow></math></span>, and <span><math><mi>A</mi></math></span>, <span><math><mi>B</mi></math></span> are either zero, or one of them is a nonzero homogeneous polynomial of degree 3, <span><math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mrow><mo>{</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow
{"title":"Limit cycles of a class of hybrid piecewise differential systems with a discontinuity line of L shape","authors":"Marly Tatiana Anacona Cabrera , Gerardo Anacona Erazo , Jaume Llibre","doi":"10.1016/j.nonrwa.2025.104492","DOIUrl":"10.1016/j.nonrwa.2025.104492","url":null,"abstract":"<div><div>In this work we study a class of discontinuous hybrid piecewise differential systems formed by two Hamiltonian systems that we named piecewise hybrid Hamiltonian systems. More precisely, we consider the differential systems with Hamiltonian functions <span><span><span><span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>y</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>5</mn></mrow></msub><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>A</mi><mo>,</mo><mtext>if</mtext><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span></span><span><span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>y</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>4</mn></mrow></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>5</mn></mrow></msub><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>B</mi><mo>,</mo><mtext>if</mtext><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>Σ</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math></span></span></span></span>with reset maps <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mi>x</mi></mrow></math></span> on <span><math><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>r</mi><mi>y</mi></mrow></math></span> on <span><math><mrow><mi>y</mi><mo>≥</mo><mn>0</mn></mrow></math></span> for <span><math><mrow><mn>0</mn><mo><</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo><</mo><mn>1</mn></mrow></math></span>, and <span><math><mi>A</mi></math></span>, <span><math><mi>B</mi></math></span> are either zero, or one of them is a nonzero homogeneous polynomial of degree 3, <span><math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mrow><mo>{</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104492"},"PeriodicalIF":1.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157517","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 : 2025-09-25DOI: 10.1016/j.nonrwa.2025.104508
Eduardo Casas , Karl Kunisch , Fredi Tröltzsch
The value function for an infinite horizon tracking type optimal control problem with semilinear parabolic equation is investigated. In view of a possible nonconvexity of the optimal control problem, a local version of the value function is considered. Its differentiability is proved for initial data in a neighborhood around the nominal initial value, provided a second order sufficient optimality condition is fulfilled for the nominal locally optimal control. Based on the differentiability of the value function, a Hamilton-Jacobi-Bellman equation is derived.
{"title":"On the value function for optimal control of semilinear parabolic equations","authors":"Eduardo Casas , Karl Kunisch , Fredi Tröltzsch","doi":"10.1016/j.nonrwa.2025.104508","DOIUrl":"10.1016/j.nonrwa.2025.104508","url":null,"abstract":"<div><div>The value function for an infinite horizon tracking type optimal control problem with semilinear parabolic equation is investigated. In view of a possible nonconvexity of the optimal control problem, a local version of the value function is considered. Its differentiability is proved for initial data in a neighborhood around the nominal initial value, provided a second order sufficient optimality condition is fulfilled for the nominal locally optimal control. Based on the differentiability of the value function, a Hamilton-Jacobi-Bellman equation is derived.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104508"},"PeriodicalIF":1.8,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157632","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 : 2025-09-24DOI: 10.1016/j.nonrwa.2025.104481
Charles M. Elliott, Thomas Sales
We consider the existence of suitable weak solutions to the Cahn-Hilliard equation with a non-constant (degenerate) mobility on a class of evolving surfaces. We also show weak-strong uniqueness for the case of a positive mobility function, and under some further assumptions on the initial data we show uniqueness for a class of strong solutions for a degenerate mobility function.
{"title":"The evolving surface Cahn–Hilliard equation with a degenerate mobility","authors":"Charles M. Elliott, Thomas Sales","doi":"10.1016/j.nonrwa.2025.104481","DOIUrl":"10.1016/j.nonrwa.2025.104481","url":null,"abstract":"<div><div>We consider the existence of suitable weak solutions to the Cahn-Hilliard equation with a non-constant (degenerate) mobility on a class of evolving surfaces. We also show weak-strong uniqueness for the case of a positive mobility function, and under some further assumptions on the initial data we show uniqueness for a class of strong solutions for a degenerate mobility function.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104481"},"PeriodicalIF":1.8,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117798","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":"2025-09-23","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 : 2025-09-23DOI: 10.1016/j.nonrwa.2025.104507
Yazhou Chen , Yunkun Chen , Xue Wang
The three-dimensional (3D) full compressible magnetohydrodynamic system is studied in a general bounded domain with slip boundary condition for the velocity filed, adiabatic condition for the temperature and perfect conduction for the magnetic field. For the regular initial data with small energy but possibly large oscillations, the global existence of classical and weak solution as well as the exponential decay rate to the initial-boundary-value problem of this system is obtained. In particular, the density and temperature of such a classical solution are both allowed to vanish initially. Moreover, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). Some new observations and useful estimates are developed to overcome the difficulties caused by the slip boundary conditions.
{"title":"Global well-posedness of full compressible magnetohydrodynamic system in 3D bounded domains with large oscillations and vacuum","authors":"Yazhou Chen , Yunkun Chen , Xue Wang","doi":"10.1016/j.nonrwa.2025.104507","DOIUrl":"10.1016/j.nonrwa.2025.104507","url":null,"abstract":"<div><div>The three-dimensional (3D) full compressible magnetohydrodynamic system is studied in a general bounded domain with slip boundary condition for the velocity filed, adiabatic condition for the temperature and perfect conduction for the magnetic field. For the regular initial data with small energy but possibly large oscillations, the global existence of classical and weak solution as well as the exponential decay rate to the initial-boundary-value problem of this system is obtained. In particular, the density and temperature of such a classical solution are both allowed to vanish initially. Moreover, it is also shown that for the classical solutions, the oscillation of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). Some new observations and useful estimates are developed to overcome the difficulties caused by the slip boundary conditions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104507"},"PeriodicalIF":1.8,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117682","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 : 2025-09-21DOI: 10.1016/j.nonrwa.2025.104503
Anna Cima, Armengol Gasull, Francesc Mañosas
It is well known that the critical points of planar polynomial Hamiltonian vector fields are either centers or points with an even number of hyperbolic sectors. We give a sharp upper bound of the number of centers that these systems can have in terms of the degrees of their components. We also prove that generically the critical points at infinity of their Poincaré compactification are either nodes or have indices or 1 and have at most two sectors: both hyperbolic, both elliptic or one of each type. These characteristics are no more true in the non generic situation. Although these results are known we revisit their proofs. The new proofs are shorter and based on a new approach.
{"title":"On the critical points of planar polynomial Hamiltonian systems","authors":"Anna Cima, Armengol Gasull, Francesc Mañosas","doi":"10.1016/j.nonrwa.2025.104503","DOIUrl":"10.1016/j.nonrwa.2025.104503","url":null,"abstract":"<div><div>It is well known that the critical points of planar polynomial Hamiltonian vector fields are either centers or points with an even number of hyperbolic sectors. We give a sharp upper bound of the number of centers that these systems can have in terms of the degrees of their components. We also prove that generically the critical points at infinity of their Poincaré compactification are either nodes or have indices <span><math><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></math></span> or 1 and have at most two sectors: both hyperbolic, both elliptic or one of each type. These characteristics are no more true in the non generic situation. Although these results are known we revisit their proofs. The new proofs are shorter and based on a new approach.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104503"},"PeriodicalIF":1.8,"publicationDate":"2025-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099838","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 : 2025-09-19DOI: 10.1016/j.nonrwa.2025.104510
Hyeong-Ohk Bae , Jörg Wolf
In the present paper we consider a weak solution to the equations of shear thickening incompressible fluid. We prove that under a Serrin-type condition imposed on the velocity field , the field enjoys a higher integrability properties, which ensures that is strong. In particular, we prove that for powers law any weak solution is strong.
{"title":"Serrin-type condition for weak solutions to the shear thickening non-Newtonian fluid","authors":"Hyeong-Ohk Bae , Jörg Wolf","doi":"10.1016/j.nonrwa.2025.104510","DOIUrl":"10.1016/j.nonrwa.2025.104510","url":null,"abstract":"<div><div>In the present paper we consider a weak solution to the equations of shear thickening incompressible fluid. We prove that under a Serrin-type condition imposed on the velocity field <span><math><mi>u</mi></math></span>, the field enjoys a higher integrability properties, which ensures that <span><math><mi>u</mi></math></span> is strong. In particular, we prove that for powers law <span><math><mrow><mi>q</mi><mo>≥</mo><mfrac><mn>11</mn><mn>5</mn></mfrac></mrow></math></span> any weak solution is strong.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104510"},"PeriodicalIF":1.8,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099443","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.2025.104505
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":"2025-09-17","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}
This article addresses the homogenization of the heat equation involving the -Laplacian in non-cylindrical domains with an evolving oscillating boundary. A change of coordinates is employed to transform the heat equations with -Laplacian into parabolic -Laplacian equations featuring oscillating coefficients in a reference domain. One novelty of this article is that the equation in the reference domain consists of an oscillating coefficient matrix in the nonlinear component, specifically . The existence and uniqueness of solutions are demonstrated in the reference domain through a non-trivial Galerkin approximation, accompanied by a significant -uniform estimate. On the other hand, a modified two-scale convergence method is employed to derive the two-scale homogenized problem. Furthermore, an explicit solution to the nonlinear cell problem is constructed. This solution is employed to drive the effective equation within the reference domain and corrector result, identified as a transformed effective problem of the heat equation with -Laplacian in a non-cylindrical domain featuring an effective evolving boundary.
{"title":"A p-Laplacian heat equation in a non-cylindrical domain with an oscillating boundary: A homogenization process","authors":"Akambadath Keerthiyil Nandakumaran , Sankar Kasinathan","doi":"10.1016/j.nonrwa.2025.104501","DOIUrl":"10.1016/j.nonrwa.2025.104501","url":null,"abstract":"<div><div>This article addresses the homogenization of the heat equation involving the <span><math><mi>p</mi></math></span>-Laplacian in non-cylindrical domains with an evolving oscillating boundary. A change of coordinates is employed to transform the heat equations with <span><math><mi>p</mi></math></span>-Laplacian into parabolic <span><math><mi>p</mi></math></span>-Laplacian equations featuring oscillating coefficients in a reference domain. One novelty of this article is that the equation in the reference domain consists of an oscillating coefficient matrix in the nonlinear component, specifically <span><math><msup><mrow><mo>|</mo><msubsup><mi>M</mi><mrow><mrow><mi>ε</mi></mrow></mrow><mrow><mi>t</mi><mi>r</mi></mrow></msubsup><mi>∇</mi><msub><mi>U</mi><mrow><mi>ε</mi></mrow></msub><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span>. The existence and uniqueness of solutions are demonstrated in the reference domain through a non-trivial Galerkin approximation, accompanied by a significant <span><math><mrow><mi>ε</mi></mrow></math></span>-uniform estimate. On the other hand, a modified two-scale convergence method is employed to derive the two-scale homogenized problem. Furthermore, an explicit solution to the nonlinear cell problem is constructed. This solution is employed to drive the effective equation within the reference domain and corrector result, identified as a transformed effective problem of the heat equation with <span><math><mi>p</mi></math></span>-Laplacian in a non-cylindrical domain featuring an effective evolving boundary.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104501"},"PeriodicalIF":1.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099835","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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