{"title":"PRM volume 153 issue 5 Cover and Front matter","authors":"","doi":"10.1017/prm.2023.71","DOIUrl":"https://doi.org/10.1017/prm.2023.71","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"1 1","pages":"f1 - f2"},"PeriodicalIF":1.3,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74428970","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}
{"title":"PRM volume 153 issue 5 Cover and Back matter","authors":"","doi":"10.1017/prm.2023.72","DOIUrl":"https://doi.org/10.1017/prm.2023.72","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"36 1","pages":"b1 - b2"},"PeriodicalIF":1.3,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74722916","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}
Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.
{"title":"Stability properties of multidimensional symmetric hyperbolic systems with damping, differential constraints and delay","authors":"Gilbert Peralta","doi":"10.1017/prm.2023.93","DOIUrl":"https://doi.org/10.1017/prm.2023.93","url":null,"abstract":"Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79622859","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}
We consider the system of partial differential equations [ begin{cases} eta_t - alpha u_{xxx} - beta eta_{xx} = 0 u_t + eta_x + beta u_{xx} = 0 end{cases} ] on bounded domains, known in the literature as the Whitham–Broer–Kaup system. The well-posedness of the problem, under suitable boundary conditions, is addressed, and it is shown to depend on the sign of the number [ varkappa=alpha-beta^2. ] In particular, existence and uniqueness occur if and only if $varkappa >0$ . In which case, an explicit representation for the solutions is given. Nonetheless, for the case $varkappa leq 0$ we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.
{"title":"On the linearized Whitham–Broer–Kaup system on bounded domains","authors":"L. Liverani, Y. Mammeri, V. Pata, R. Quintanilla","doi":"10.1017/prm.2023.85","DOIUrl":"https://doi.org/10.1017/prm.2023.85","url":null,"abstract":"We consider the system of partial differential equations\u0000\u0000 \u0000 [ begin{cases} eta_t - alpha u_{xxx} - beta eta_{xx} = 0 u_t + eta_x + beta u_{xx} = 0 end{cases} ]\u0000 \u0000 \u0000 on bounded domains, known in the literature as the Whitham–Broer–Kaup system. The well-posedness of the problem, under suitable boundary conditions, is addressed, and it is shown to depend on the sign of the number\u0000\u0000 \u0000 [ varkappa=alpha-beta^2. ]\u0000 \u0000 \u0000 In particular, existence and uniqueness occur if and only if \u0000 \u0000 $varkappa >0$\u0000 \u0000 \u0000 . In which case, an explicit representation for the solutions is given. Nonetheless, for the case \u0000 \u0000 $varkappa leq 0$\u0000 \u0000 \u0000 we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"54 1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80931429","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}
In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$ Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.
{"title":"A weighted Trudinger–Moser inequalities and applications to some weighted Laplacian equation in with new exponential growth conditions","authors":"S. Aouaoui","doi":"10.1017/prm.2023.86","DOIUrl":"https://doi.org/10.1017/prm.2023.86","url":null,"abstract":"In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted \u0000 \u0000 $(N,q)-$\u0000 \u0000 \u0000 Laplacian operator where \u0000 \u0000 $1 < q < N$\u0000 \u0000 \u0000 and a nonlinearities enjoying a new type of exponential growth condition at infinity.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"35 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85724963","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}
For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.
{"title":"Periodic and solitary waves in a Korteweg–de Vries equation with delay","authors":"Qi Qiao, Shuling Yan, Xiang Zhang","doi":"10.1017/prm.2023.88","DOIUrl":"https://doi.org/10.1017/prm.2023.88","url":null,"abstract":"<p>For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":" 27","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492524","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}
In this paper, we give necessary conditions for an $N$ -expansive homeomorphism of a compact metric space to be nonchaotic in the Li–Yorke sense. As application we give a partial answer to a conjecture in [2].
{"title":"Nonchaotic N-expansive homeomorphisms","authors":"W. Jung, C. Morales","doi":"10.1017/prm.2023.76","DOIUrl":"https://doi.org/10.1017/prm.2023.76","url":null,"abstract":"In this paper, we give necessary conditions for an \u0000 \u0000 $N$\u0000 \u0000 \u0000 -expansive homeomorphism of a compact metric space to be nonchaotic in the Li–Yorke sense. As application we give a partial answer to a conjecture in [2].","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89492470","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}
In this paper, we study existence of rotating periodic solutions for p-Laplacian differential systems. We first build a new continuation theorem by topological degree, and then obtain the existence of rotating periodic solutions for two kinds of p-Laplacian differential systems via this continuation theorem, extend some existing relevant results.
{"title":"Rotating periodic solutions for p-Laplacian differential systems","authors":"Tiefeng Ye, Wenbin Liu, Tengfei Shen","doi":"10.1017/prm.2023.83","DOIUrl":"https://doi.org/10.1017/prm.2023.83","url":null,"abstract":"In this paper, we study existence of rotating periodic solutions for p-Laplacian differential systems. We first build a new continuation theorem by topological degree, and then obtain the existence of rotating periodic solutions for two kinds of p-Laplacian differential systems via this continuation theorem, extend some existing relevant results.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75150383","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}
We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $operatorname K$ -theory of stable $infty$ -categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $operatorname K$ -theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $operatorname K$ -theory.
{"title":"The localisation theorem for the K-theory of stable ∞-categories","authors":"F. Hebestreit, Andrea Lachmann, W. Steimle","doi":"10.1017/prm.2023.35","DOIUrl":"https://doi.org/10.1017/prm.2023.35","url":null,"abstract":"We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic \u0000 \u0000 $operatorname K$\u0000 \u0000 \u0000 -theory of stable \u0000 \u0000 $infty$\u0000 \u0000 \u0000 -categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of \u0000 \u0000 $operatorname K$\u0000 \u0000 \u0000 -theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of \u0000 \u0000 $operatorname K$\u0000 \u0000 \u0000 -theory.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72910843","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}
Bo Li, Jinxia Li, Qingze Lin, Bolin Ma, Tianjun Shen
Let $mathcal {M}$ be an Ahlfors $n$ -regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : mathcal {M} to mathbb {R}$ by a Carleson measure condition of its $sigma$ -harmonic extension $U:mathcal {M}times mathbb {R}_+ to mathbb {R}$ . This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.
{"title":"A revisit to “On BMO and Carleson measures on Riemannian manifolds”","authors":"Bo Li, Jinxia Li, Qingze Lin, Bolin Ma, Tianjun Shen","doi":"10.1017/prm.2023.58","DOIUrl":"https://doi.org/10.1017/prm.2023.58","url":null,"abstract":"Let $mathcal {M}$ be an Ahlfors $n$ -regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : mathcal {M} to mathbb {R}$ by a Carleson measure condition of its $sigma$ -harmonic extension $U:mathcal {M}times mathbb {R}_+ to mathbb {R}$ . This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75989665","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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