New Hardy type inequalities in sectorial area and as a limit in an exterior of a ball are proved. Sharpness of the inequalities is shown as well.
证明了扇形面积上的新Hardy型不等式和球外的极限不等式。不等式的尖锐性也得到了体现。
{"title":"Sharp Hardy inequalities in an exterior of a ball","authors":"N. Kutev, T. Rangelov","doi":"10.1063/5.0040127","DOIUrl":"https://doi.org/10.1063/5.0040127","url":null,"abstract":"New Hardy type inequalities in sectorial area and as a limit in an exterior of a ball are proved. Sharpness of the inequalities is shown as well.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91229717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-29DOI: 10.1007/s10884-020-09917-5
P. Quittner
{"title":"An Optimal Liouville Theorem for the Linear Heat Equation with a Nonlinear Boundary Condition","authors":"P. Quittner","doi":"10.1007/s10884-020-09917-5","DOIUrl":"https://doi.org/10.1007/s10884-020-09917-5","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"1983 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87802384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study a ratio-dependent predator-prey model with a free boundary causing by both prey and predator over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy, namely, as t goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and die out eventually. Then the criteria governing spreading and vanishing are obtained. Finally, when spreading occurs, we provide some estimates to the asymptotic spreading speed of h(t).
{"title":"A Free Boundary Problem with a Stefan Condition for a Ratio-dependent Predator-prey Model","authors":"Lingyu Liu","doi":"10.3934/MATH.2021711","DOIUrl":"https://doi.org/10.3934/MATH.2021711","url":null,"abstract":"In this paper we study a ratio-dependent predator-prey model with a free boundary causing by both prey and predator over a one dimensional habitat. We study the long time behaviors of the two species and prove a spreading-vanishing dichotomy, namely, as t goes to infinity, both prey and predator successfully spread to the whole space and survive in the new environment, or they spread within a bounded area and die out eventually. Then the criteria governing spreading and vanishing are obtained. Finally, when spreading occurs, we provide some estimates to the asymptotic spreading speed of h(t).","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83738686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-27DOI: 10.1142/s1664360721500077
Alejandro Gárriz, L. Ignat
In this article the authors study a non-local diffusion problem that involves three different fractional laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the article we provide results about existence of solutions and the conservation of mass. The second part is devoted to study the asymptotic behaviour of the solutions of the problem when the two domains are a ball and its complementary. Fractional Sobolev inequalities in exterior domains are also provided.
{"title":"A non-local coupling model involving three fractional Laplacians","authors":"Alejandro Gárriz, L. Ignat","doi":"10.1142/s1664360721500077","DOIUrl":"https://doi.org/10.1142/s1664360721500077","url":null,"abstract":"In this article the authors study a non-local diffusion problem that involves three different fractional laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the article we provide results about existence of solutions and the conservation of mass. The second part is devoted to study the asymptotic behaviour of the solutions of the problem when the two domains are a ball and its complementary. Fractional Sobolev inequalities in exterior domains are also provided.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90420314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-26DOI: 10.1016/J.MATPUR.2021.02.006
Laurent Lafleche, A. Vasseur, M. Vishik
{"title":"Instability for axisymmetric blow-up solutions to incompressible Euler equations","authors":"Laurent Lafleche, A. Vasseur, M. Vishik","doi":"10.1016/J.MATPUR.2021.02.006","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.02.006","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75534230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)to 0$ as $xto -infty$ and $q(x,0)to A$ as $xto +infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the "shifted step function" $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction $x/t=const$).
{"title":"Defocusing Nonlocal Nonlinear Schrödinger Equation with Step-like Boundary Conditions: Long-time Behavior for Shifted Initial Data","authors":"Yan Rybalko, D. Shepelsky","doi":"10.15407/mag16.04.418","DOIUrl":"https://doi.org/10.15407/mag16.04.418","url":null,"abstract":"The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)bar{q}(-x,t)=0 $ with a step-like initial data: $q(x,0)to 0$ as $xto -infty$ and $q(x,0)to A$ as $xto +infty$. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by $R>0$, with the initial data that can be viewed as perturbations of the \"shifted step function\" $q_{R,A}(x)$: $q_{R,A}(x)=0$ for $x R$, where $A>0$ and $R>0$ are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the $(x,t)$ plane, the number of which depends on the relationship between $A$ and $R$: for a fixed $A$, the bigger $R$, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction $x/t=const$).","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89603448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-25DOI: 10.1016/J.PHYSD.2021.132891
Mathew A. Johnson, Wesley R. Perkins
{"title":"Subharmonic dynamics of wave trains in reaction–diffusion systems","authors":"Mathew A. Johnson, Wesley R. Perkins","doi":"10.1016/J.PHYSD.2021.132891","DOIUrl":"https://doi.org/10.1016/J.PHYSD.2021.132891","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77627765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maria Colombo, Michele Coti Zelati, Klaus Widmayer
This article addresses mixing and diffusion properties of passive scalars advected by rough ($C^alpha$) shear flows. We show that in general, one cannot expect a rough shear flow to increase the rate of inviscid mixing to more than that of a smooth shear without critical points. On the other hand, diffusion may be enhanced at a much faster rate. This shows that in the setting of low regularity, the interplay between inviscid mixing properties and enhanced dissipation is more intricate, and in fact contradicts some of the natural heuristics that are valid in the smooth setting.
{"title":"Mixing and diffusion for rough shear flows.","authors":"Maria Colombo, Michele Coti Zelati, Klaus Widmayer","doi":"10.15781/83fc-j334","DOIUrl":"https://doi.org/10.15781/83fc-j334","url":null,"abstract":"This article addresses mixing and diffusion properties of passive scalars advected by rough ($C^alpha$) shear flows. We show that in general, one cannot expect a rough shear flow to increase the rate of inviscid mixing to more than that of a smooth shear without critical points. On the other hand, diffusion may be enhanced at a much faster rate. This shows that in the setting of low regularity, the interplay between inviscid mixing properties and enhanced dissipation is more intricate, and in fact contradicts some of the natural heuristics that are valid in the smooth setting.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76216033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-22DOI: 10.1142/s0218202521500287
Hai-yang Jin, Tian Xiang
In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotaxis on asymptotical behavior, we show that any global-in-time haptotaxis solution component vanishes exponentially as time approaches infinity, and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, the aforementioned critical mass blow-up phenomenon for the chemotaxis-only model is almost undestroyed even with arbitrary introduction of haptotaixs, showing negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and longtime behavior in the chemotaxis-haptotaxis model.
{"title":"Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model","authors":"Hai-yang Jin, Tian Xiang","doi":"10.1142/s0218202521500287","DOIUrl":"https://doi.org/10.1142/s0218202521500287","url":null,"abstract":"In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller-Segel chemotaxis-only model possesses a striking feature of critical mass blow-up phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model. For negligibility of haptotaxis on asymptotical behavior, we show that any global-in-time haptotaxis solution component vanishes exponentially as time approaches infinity, and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, the aforementioned critical mass blow-up phenomenon for the chemotaxis-only model is almost undestroyed even with arbitrary introduction of haptotaixs, showing negligibility of haptotaxis effect compared to chemotaxis effect in terms of boundedness, blow-up and longtime behavior in the chemotaxis-haptotaxis model.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74082902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-19DOI: 10.1016/j.matpur.2021.08.007
M. Alfaro, Léo Girardin, F. Hamel, L. Roques
{"title":"When the Allee threshold is an evolutionary trait: Persistence vs. extinction","authors":"M. Alfaro, Léo Girardin, F. Hamel, L. Roques","doi":"10.1016/j.matpur.2021.08.007","DOIUrl":"https://doi.org/10.1016/j.matpur.2021.08.007","url":null,"abstract":"","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82064996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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