Pub Date : 2025-10-07DOI: 10.1016/j.nonrwa.2025.104512
Nebi Yılmaz , Muhteşem Demir , Erhan Pişkin
This study explores a higher-order viscoelastic equation characterized by variable exponents. We demonstrate the local existence of weak solutions by imposing appropriate conditions on these variable exponents. Furthermore, we investigate the phenomenon of finite-time blow-up for solutions that begin with positive initial energy. Finally, we give a 2D numerical example for the blow up.
{"title":"Local existence and blow-up of solutions for the higher-order viscoelastic equation with general source term and variable exponents: Theoretical and numerical results","authors":"Nebi Yılmaz , Muhteşem Demir , Erhan Pişkin","doi":"10.1016/j.nonrwa.2025.104512","DOIUrl":"10.1016/j.nonrwa.2025.104512","url":null,"abstract":"<div><div>This study explores a higher-order viscoelastic equation characterized by variable exponents. We demonstrate the local existence of weak solutions by imposing appropriate conditions on these variable exponents. Furthermore, we investigate the phenomenon of finite-time blow-up for solutions that begin with positive initial energy. Finally, we give a 2D numerical example for the blow up.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"89 ","pages":"Article 104512"},"PeriodicalIF":1.8,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270702","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 : 2025-10-04DOI: 10.1016/j.nonrwa.2025.104513
Juan Límaco, João Carlos Barreira, Suerlan Silva, Luis P. Yapu
In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called leader and two controls called followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, Límaco and Viana [31] and the local controllability of the non-linear system is obtained using Liusternik’s inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem.
{"title":"Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient","authors":"Juan Límaco, João Carlos Barreira, Suerlan Silva, Luis P. Yapu","doi":"10.1016/j.nonrwa.2025.104513","DOIUrl":"10.1016/j.nonrwa.2025.104513","url":null,"abstract":"<div><div>In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called <em>leader</em> and two controls called <em>followers</em>. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, Límaco and Viana [31] and the local controllability of the non-linear system is obtained using Liusternik’s inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"89 ","pages":"Article 104513"},"PeriodicalIF":1.8,"publicationDate":"2025-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223607","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 : 2025-09-29DOI: 10.1016/j.nonrwa.2025.104493
Xuanxuan Han, Shaojie Yang
This paper is concerned with the solutions of the quadratic-cubic Camassa–Holm equation which is a model that explore the change in the physical structure of the solutions from the peakons to the bell-shaped solitary wave solutions. The first type of solutions exhibits finite time singularity in the sense of wave breaking. We perform a refined analysis based on the local structure of the dynamics to provide a condition on the initial data to guarantee wave breaking. The key feature of the method is to refine the analysis on characteristics and conserved quantities to the Riccati-type differential inequality. The other type of solutions which we study is the traveling waves, we investigate nonexistence of the Camassa–Holm-type peaked traveling wave solutions. Moreover, we discover how the symmetric structure is connected to the steady structure of solutions to the quadratic-cubic Camassa–Holm equation, and prove that the classical symmetric waves must be traveling wave solutions.
{"title":"Wave breaking and traveling waves for the quadratic-cubic Camassa–Holm equation","authors":"Xuanxuan Han, Shaojie Yang","doi":"10.1016/j.nonrwa.2025.104493","DOIUrl":"10.1016/j.nonrwa.2025.104493","url":null,"abstract":"<div><div>This paper is concerned with the solutions of the quadratic-cubic Camassa–Holm equation which is a model that explore the change in the physical structure of the solutions from the peakons to the bell-shaped solitary wave solutions. The first type of solutions exhibits finite time singularity in the sense of wave breaking. We perform a refined analysis based on the local structure of the dynamics to provide a condition on the initial data to guarantee wave breaking. The key feature of the method is to refine the analysis on characteristics and conserved quantities to the Riccati-type differential inequality. The other type of solutions which we study is the traveling waves, we investigate nonexistence of the Camassa–Holm-type peaked traveling wave solutions. Moreover, we discover how the symmetric structure is connected to the steady structure of solutions to the quadratic-cubic Camassa–Holm equation, and prove that the classical symmetric waves must be traveling wave solutions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104493"},"PeriodicalIF":1.8,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219604","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 : 2025-09-26DOI: 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}
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