Pub Date : 2025-01-21DOI: 10.1016/j.jmaa.2025.129290
Quanyong Zhao, Jinrong Wang
<div><div>In this paper, we study the following chemotaxis-growth system with density-suppressed motility<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>u</mi><mi>γ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>r</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>δ</mi><mi>w</mi><mo>+</mo><mi>u</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> under homogeneous Neumann boundary conditions in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span> with smooth boundary, where <span><math><mi>r</mi><mo>∈</mo><mi>R</mi></math></span>, <em>μ</em>, <em>δ</em>, <span><math><mi>β</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mi>α</mi><mo>></mo><mn>1</mn></math></span>, and the positive motility function <span><math><mi>γ</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span> satisfies <span><math><msup><mrow><mi>γ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>s</mi><mo>)</mo><mo>≤</mo><mn>0</mn></math></span> for all <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>. The main purpose of this paper is to establish the global boundedness and stabilization of classical solutions for such kind of chemotaxis system. More precisely, we showed that if <span><math><mi>β</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, the system possesses a globally bounded classical solution, and further, if <em>γ</em> also fulfills the additional assumption that <span><math><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>γ</mi></mrow></mfrac></math></span> is bounded on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, then for <span><math><mi>α</mi><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>, the above restriction can be optimized as <span><math><mi>β</mi><mo><</mo><mfrac><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>α</mi><mo>
{"title":"Nonlinear generation index of bounded solutions in a chemotaxis system with indirect signal, density-suppressed motility and generalized logistic source","authors":"Quanyong Zhao, Jinrong Wang","doi":"10.1016/j.jmaa.2025.129290","DOIUrl":"10.1016/j.jmaa.2025.129290","url":null,"abstract":"<div><div>In this paper, we study the following chemotaxis-growth system with density-suppressed motility<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mo>(</mo><mi>u</mi><mi>γ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>r</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><msup><mrow><mi>w</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>δ</mi><mi>w</mi><mo>+</mo><mi>u</mi><mo>,</mo></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> under homogeneous Neumann boundary conditions in a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span> with smooth boundary, where <span><math><mi>r</mi><mo>∈</mo><mi>R</mi></math></span>, <em>μ</em>, <em>δ</em>, <span><math><mi>β</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mi>α</mi><mo>></mo><mn>1</mn></math></span>, and the positive motility function <span><math><mi>γ</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span> satisfies <span><math><msup><mrow><mi>γ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>s</mi><mo>)</mo><mo>≤</mo><mn>0</mn></math></span> for all <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>. The main purpose of this paper is to establish the global boundedness and stabilization of classical solutions for such kind of chemotaxis system. More precisely, we showed that if <span><math><mi>β</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>α</mi></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, the system possesses a globally bounded classical solution, and further, if <em>γ</em> also fulfills the additional assumption that <span><math><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mi>γ</mi></mrow></mfrac></math></span> is bounded on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, then for <span><math><mi>α</mi><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>, the above restriction can be optimized as <span><math><mi>β</mi><mo><</mo><mfrac><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>α</mi><mo>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129290"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166892","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 : 2025-01-21DOI: 10.1016/j.jmaa.2025.129284
Alexandre N. Carvalho , Jacson Simsen , Mariza S. Simsen
In this work we consider a family of quasilinear equations with variable exponents (-Laplacian) and perturbations which are not globally Lipschitz. We prove existence of global solutions, existence of global attractors and we provide conditions on the data in order that the associated semilinear equation () commands the asymptotic dynamics of the family of problems when the exponents are sufficiently close to 2 (uniformly in x) by showing the continuity of the flows and the upper semicontinuity of the global attractors.
{"title":"Attractors for parabolic problems with p(x)-Laplacian: Bounds, continuity of the flow and robustness","authors":"Alexandre N. Carvalho , Jacson Simsen , Mariza S. Simsen","doi":"10.1016/j.jmaa.2025.129284","DOIUrl":"10.1016/j.jmaa.2025.129284","url":null,"abstract":"<div><div>In this work we consider a family of quasilinear equations with variable exponents (<span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>-Laplacian) and perturbations which are not globally Lipschitz. We prove existence of global solutions, existence of global attractors and we provide conditions on the data in order that the associated semilinear equation (<span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≡</mo><mn>2</mn></math></span>) commands the asymptotic dynamics of the family of problems when the exponents are sufficiently close to 2 (uniformly in <em>x</em>) by showing the continuity of the flows and the upper semicontinuity of the global attractors.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129284"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166876","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 : 2025-01-21DOI: 10.1016/j.jmaa.2025.129292
Fábio R. dos Santos, Joicy P. Cruz
A Simons-type equation for hypersurfaces in a product space , where is a Riemannian manifold with constant sectional curvature κ, is derived. Several applications are presented, including those for stochastically complete hypersurfaces with zero mean curvature and nonzero constant mean curvature in these spaces.
{"title":"On the stochastically complete hypersurfaces in the product spaces Mn(κ)×R","authors":"Fábio R. dos Santos, Joicy P. Cruz","doi":"10.1016/j.jmaa.2025.129292","DOIUrl":"10.1016/j.jmaa.2025.129292","url":null,"abstract":"<div><div>A Simons-type equation for hypersurfaces in a product space <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>κ</mi><mo>)</mo><mo>×</mo><mi>R</mi></math></span>, where <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>κ</mi><mo>)</mo></math></span> is a Riemannian manifold with constant sectional curvature <em>κ</em>, is derived. Several applications are presented, including those for stochastically complete hypersurfaces with zero mean curvature and nonzero constant mean curvature in these spaces.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"546 2","pages":"Article 129292"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165799","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 : 2025-01-21DOI: 10.1016/j.jmaa.2025.129291
Arunmaran Mahenthiram
A Brownian particle released from a point in a two-dimensional region Ω will move around randomly until it eventually hits the boundary of Ω. We are interested in the probability that it hits the boundary somewhere within distance r of the starting point , for each value of r. Putting together these probabilities for all values of r gives a function called the harmonic-measure distribution function or h-function of Ω with respect to . This h-function encodes information about the shape of the boundary of Ω. In this paper, we compute the h-functions for some multiply connected planar regions whose boundary consists of collinear unequal slits or dissimilar discs. The key tool we use for computing these h-functions is a special function, called the Schottky-Klein prime function. Furthermore, we have validated our results by simulating the random motion of Brownian particles in the regions mentioned above.
{"title":"Computing harmonic-measure distribution functions of some multiply connected unbounded planar domains","authors":"Arunmaran Mahenthiram","doi":"10.1016/j.jmaa.2025.129291","DOIUrl":"10.1016/j.jmaa.2025.129291","url":null,"abstract":"<div><div>A Brownian particle released from a point <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> in a two-dimensional region Ω will move around randomly until it eventually hits the boundary of Ω. We are interested in the probability that it hits the boundary somewhere within distance <em>r</em> of the starting point <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, for each value of <em>r</em>. Putting together these probabilities for all values of <em>r</em> gives a function <span><math><mi>h</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> called the harmonic-measure distribution function or <em>h</em>-function of Ω with respect to <span><math><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. This <em>h</em>-function encodes information about the shape of the boundary of Ω. In this paper, we compute the <em>h</em>-functions for some multiply connected planar regions whose boundary consists of collinear unequal slits or dissimilar discs. The key tool we use for computing these <em>h</em>-functions is a special function, called the <em>Schottky-Klein</em> prime function. Furthermore, we have validated our results by simulating the random motion of Brownian particles in the regions mentioned above.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129291"},"PeriodicalIF":1.2,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166390","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 : 2025-01-20DOI: 10.1016/j.jmaa.2025.129283
V.J. Ervin
In this paper we analyze the existence, uniqueness and regularity of the solution to the generalized, variable diffusivity, fractional Laplace equation on the unit disk in . For α the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, , satisfying , for all , , with , the problem has a unique solution. The regularity of the solution is given in an appropriately weighted Sobolev space in terms of the regularity of the right hand side function and .
{"title":"A variable diffusivity fractional Laplacian","authors":"V.J. Ervin","doi":"10.1016/j.jmaa.2025.129283","DOIUrl":"10.1016/j.jmaa.2025.129283","url":null,"abstract":"<div><div>In this paper we analyze the existence, uniqueness and regularity of the solution to the generalized, variable diffusivity, fractional Laplace equation on the unit disk in <span><math><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mn>2</mn></mrow></msup></math></span>. For <em>α</em> the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, <span><math><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, satisfying <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi></mrow></msub><mspace></mspace><msup><mrow><mi>v</mi></mrow><mrow><mi>T</mi></mrow></msup><mspace></mspace><mi>v</mi><mspace></mspace><mo>≤</mo><mspace></mspace><msup><mrow><mi>v</mi></mrow><mrow><mi>T</mi></mrow></msup><mspace></mspace><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace><mi>v</mi><mspace></mspace><mo>≤</mo><mspace></mspace><msub><mrow><mi>λ</mi></mrow><mrow><mi>M</mi></mrow></msub><mspace></mspace><msup><mrow><mi>v</mi></mrow><mrow><mi>T</mi></mrow></msup><mspace></mspace><mi>v</mi></math></span>, for all <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mn>2</mn></mrow></msup></math></span>, <span><math><mi>x</mi><mo>∈</mo><mi>Ω</mi></math></span>, with <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>M</mi></mrow></msub><mspace></mspace><mo><</mo><mspace></mspace><mfrac><mrow><msqrt><mrow><mi>α</mi><mspace></mspace><mo>(</mo><mn>2</mn><mo>+</mo><mi>α</mi><mo>)</mo></mrow></msqrt></mrow><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></mfrac><mspace></mspace><msub><mrow><mi>λ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>, the problem has a unique solution. The regularity of the solution is given in an appropriately weighted Sobolev space in terms of the regularity of the right hand side function and <span><math><mi>K</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129283"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-20DOI: 10.1016/j.jmaa.2025.129281
Lucian Beznea , Oana Lupaşcu-Stamate , Alexandra Teodor
We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the non-local branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.
{"title":"Nonlinear Dirichlet problem of non-local branching processes","authors":"Lucian Beznea , Oana Lupaşcu-Stamate , Alexandra Teodor","doi":"10.1016/j.jmaa.2025.129281","DOIUrl":"10.1016/j.jmaa.2025.129281","url":null,"abstract":"<div><div>We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the non-local branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129281"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-20DOI: 10.1016/j.jmaa.2025.129289
Kotaro Hisa
In this paper we obtain necessary conditions on the initial value for the solvability of the Cauchy problem for semilinear heat equations. These necessary conditions were already obtained in the framework of integral solutions, but not in that of very weak ones. We establish a new proof method, which can derive the desired conditions in the framework of very weak solutions. In particular, since any integral solution is a very weak solution, our conditions are more general.
{"title":"Necessary conditions for the solvability of fractional semilinear heat equations in the very weak framework","authors":"Kotaro Hisa","doi":"10.1016/j.jmaa.2025.129289","DOIUrl":"10.1016/j.jmaa.2025.129289","url":null,"abstract":"<div><div>In this paper we obtain necessary conditions on the initial value for the solvability of the Cauchy problem for semilinear heat equations. These necessary conditions were already obtained in the framework of integral solutions, but not in that of very weak ones. We establish a new proof method, which can derive the desired conditions in the framework of very weak solutions. In particular, since any integral solution is a very weak solution, our conditions are more general.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129289"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166875","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 : 2025-01-20DOI: 10.1016/j.jmaa.2025.129288
Dongsheng Li, Yasheng Lyu
This paper is concerned with the solvability of the Dirichlet problem of mixed Hessian quotient equations, which precisely are which include k-Hessian equation, Hessian quotient equation and mixed k-Hessian equation. We establish the existence of global solution under degenerate case, and furthermore the existence of global solution under nondegenerate case. The uniqueness is also obtained.
{"title":"The Dirichlet problem for mixed Hessian quotient equations","authors":"Dongsheng Li, Yasheng Lyu","doi":"10.1016/j.jmaa.2025.129288","DOIUrl":"10.1016/j.jmaa.2025.129288","url":null,"abstract":"<div><div>This paper is concerned with the solvability of the Dirichlet problem of mixed Hessian quotient equations, which precisely are<span><span><span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mrow><mo>[</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></munderover><msubsup><mrow><mi>l</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow><mrow><mi>k</mi><mo>−</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mfrac><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mo>∀</mo><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>d</mi><mo>,</mo></math></span></span></span> which include <em>k</em>-Hessian equation, Hessian quotient equation and mixed <em>k</em>-Hessian equation. We establish the existence of global <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> solution under degenerate case, and furthermore the existence of global <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> solution under nondegenerate case. The uniqueness is also obtained.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129288"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166387","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 : 2025-01-20DOI: 10.1016/j.jmaa.2025.129271
Dohoon Choi , Subong Lim
<div><div>For a nonzero divisor <span><math><mi>D</mi><mo>:</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> with <span><math><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo>></mo><mn>0</mn></math></span>, let <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo><mo>,</mo><mi>D</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> be the space of meromorphic modular forms <em>f</em> of integral weight <em>k</em> on <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> such that <em>f</em> is holomorphic except at <span><math><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> and that the order of pole of <em>f</em> at each <span><math><mi>Q</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> is less than or equal to <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>Q</mi></mrow></msub></math></span>. In this paper, we give an isomorphism between <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo><mo>,</mo><mi>D</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> and the first cohomology group with a certain coefficient module <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> when <em>k</em> is a negative even integer. More generally, by considering another coefficient module <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>w</mi><mi>e</mi><mi>a</mi><mi>k</mi></mrow></msubsup></math></span>, we prove that there exists an isomorphism between <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>w</mi><mi>e</mi><mi>a</mi><mi>k</mi></mrow></msubsup><mo>)</mo></math
{"title":"Group cohomology for modular forms with singularities","authors":"Dohoon Choi , Subong Lim","doi":"10.1016/j.jmaa.2025.129271","DOIUrl":"10.1016/j.jmaa.2025.129271","url":null,"abstract":"<div><div>For a nonzero divisor <span><math><mi>D</mi><mo>:</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> with <span><math><msub><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mo>></mo><mn>0</mn></math></span>, let <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo><mo>,</mo><mi>D</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> be the space of meromorphic modular forms <em>f</em> of integral weight <em>k</em> on <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> such that <em>f</em> is holomorphic except at <span><math><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> and that the order of pole of <em>f</em> at each <span><math><mi>Q</mi><mo>∈</mo><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> is less than or equal to <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>Q</mi></mrow></msub></math></span>. In this paper, we give an isomorphism between <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo><mo>,</mo><mi>D</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> and the first cohomology group with a certain coefficient module <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> when <em>k</em> is a negative even integer. More generally, by considering another coefficient module <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>w</mi><mi>e</mi><mi>a</mi><mi>k</mi></mrow></msubsup></math></span>, we prove that there exists an isomorphism between <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo>!</mo></mrow></msubsup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>w</mi><mi>e</mi><mi>a</mi><mi>k</mi></mrow></msubsup><mo>)</mo></math","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"546 2","pages":"Article 129271"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165893","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 : 2025-01-20DOI: 10.1016/j.jmaa.2025.129287
Antonio C. Fernandes , Claudio Vidal
We consider planar central configurations of the κn-vortices problem consisting of κ groups of regular n-gons of equal vorticities, called -crown, or equivalently, we study the existence of periodic solutions, called relative equilibrium, for which the vortices rigidly rotate around the center of vorticity, with angular velocity . We derive the equations of central configurations for the general -crown. Next, we give a necessary condition for a -crown: either the rings are nested (the vertices of the two n-gons are aligned) or they must be rotated by an angle (twisted case). After that, we are able to give the exact number of central configurations in function of the ratio of vorticities. More precisely, we show that in the nested case there are two central configurations when the ratio of vorticity is positive, while for a negative ratio of vorticity there exists a unique central configuration for an appropriate radius. For the twisted case, it is observed that the study depends on the number of vortices in each n-gon and the admissible ratio of vorticities must be in an appropriate interval. Our arguments are analytic and differ significantly from the Newtonian case.
{"title":"Crown relative equilibria for the vortex problem","authors":"Antonio C. Fernandes , Claudio Vidal","doi":"10.1016/j.jmaa.2025.129287","DOIUrl":"10.1016/j.jmaa.2025.129287","url":null,"abstract":"<div><div>We consider planar central configurations of the <em>κn</em>-vortices problem consisting of <em>κ</em> groups of regular <em>n</em>-gons of equal vorticities, called <span><math><mo>(</mo><mi>κ</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-crown, or equivalently, we study the existence of periodic solutions, called relative equilibrium, for which the vortices rigidly rotate around the center of vorticity, with angular velocity <span><math><mi>λ</mi><mo>≠</mo><mn>0</mn></math></span>. We derive the equations of central configurations for the general <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-crown. Next, we give a necessary condition for a <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-crown: either the rings are nested (the vertices of the two <em>n</em>-gons are aligned) or they must be rotated by an angle <span><math><mi>π</mi><mo>/</mo><mi>n</mi></math></span> (twisted case). After that, we are able to give the exact number of central configurations in function of the ratio of vorticities. More precisely, we show that in the nested case there are two central configurations when the ratio of vorticity is positive, while for a negative ratio of vorticity there exists a unique central configuration for an appropriate radius. For the twisted case, it is observed that the study depends on the number of vortices in each <em>n</em>-gon and the admissible ratio of vorticities must be in an appropriate interval. Our arguments are analytic and differ significantly from the Newtonian case.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"547 1","pages":"Article 129287"},"PeriodicalIF":1.2,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166873","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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