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Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach 通过全局分岔法实现具有一般非线性的薛定谔方程在有界域中的归一化解
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10299
Wei Ji
{"title":"Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach","authors":"Wei Ji","doi":"arxiv-2409.10299","DOIUrl":"https://doi.org/arxiv-2409.10299","url":null,"abstract":"We obtain the existence, nonexistence and multiplicity of positive solutions\u0000with prescribed mass for nonlinear Schr\"{o}dinger equations in bounded domains\u0000via a global bifurcation approach. The nonlinearities in this paper can be mass\u0000supercritical, critical, subcritical or some mixes of these cases, and the\u0000equation can be autonomous or non-autonomous. This generalizes a result in\u0000Noris, Tavares and Verzini [emph{Anal. PDE}, 7 (8) (2014) 1807-1838], where\u0000the equation is autonomous with homogeneous nonlinearities. Besides, we have\u0000proven some orbital stability or instability results.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260672","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}
引用次数: 0
Existence results for Kazdan-Warner type equations on graphs 图上卡兹丹-瓦纳型方程的存在性结果
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10181
Pengxiu Yu
{"title":"Existence results for Kazdan-Warner type equations on graphs","authors":"Pengxiu Yu","doi":"arxiv-2409.10181","DOIUrl":"https://doi.org/arxiv-2409.10181","url":null,"abstract":"In this paper, motivated by the work of Huang-Lin-Yau (Commun. Math. Phys.\u00002020), Sun-Wang (Adv. Math. 2022) and Li-Sun-Yang (Calc. Var. Partial\u0000Differential Equations 2024), we investigate the existence of Kazdan-Warner\u0000type equations on a finite connected graph, based on the theory of Brouwer\u0000degree. Specifically, we consider the equation begin{equation*} -Delta\u0000u=h(x)f(u)-c, end{equation*} where $h$ is a real-valued function defined on\u0000the vertex set $V$, $cinmathbb{R}$ and begin{equation*} f(u)=\u0000left(1-displaystylefrac{1}{1+u^{2n}}right)e^u end{equation*} with $nin\u0000mathbb{N}^*$. Different from the previous studies, the main difficulty in this\u0000paper is to show that the corresponding equation has only three constant\u0000solutions, based on delicate analysis and the connectivity of graphs, which\u0000have not been extensively explored in previous literature.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"207 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260673","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}
引用次数: 0
Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition 一类具有诺伊曼条件的非局部演化方程的上半连续性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10065
Flank D. M. Bezerra, Silvia Sastre-Gomez, Severino H. da Silva
{"title":"Upper semicontinuity for a class of nonlocal evolution equations with Neumann condition","authors":"Flank D. M. Bezerra, Silvia Sastre-Gomez, Severino H. da Silva","doi":"arxiv-2409.10065","DOIUrl":"https://doi.org/arxiv-2409.10065","url":null,"abstract":"In this paper we consider the following nonlocal autonomous evolution\u0000equation in a bounded domain $Omega$ in $mathbb{R}^N$ [ partial_t u(x,t) =-\u0000h(x)u(x,t) + g Big(int_{Omega} J(x,y)u(y,t)dy Big) +f(x,u(x,t)) ] where\u0000$hin W^{1,infty}(Omega)$, $g: mathbb{R} to mathbb{R}$ and\u0000$f:mathbb{R}^Ntimesmathbb{R} to mathbb{R}$ are continuously differentiable\u0000function, and $J$ is a symmetric kernel; that is, $J(x,y)=J(y,x)$ for any\u0000$x,yinmathbb{R}^N$. Under additional suitable assumptions on $f$ and $g$, we\u0000study the asymptotic dynamics of the initial value problem associated to this\u0000equation in a suitable phase spaces. More precisely, we prove the existence,\u0000and upper semicontinuity of compact global attractors with respect to kernel\u0000$J$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260681","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}
引用次数: 0
Asymptotic stability of the composite wave of rarefaction wave and contact wave to nonlinear viscoelasticity model with non-convex flux 具有非凸通量的非线性粘弹性模型的稀释波和接触波复合波的渐近稳定性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10125
Zhenhua Guo, Meichen Hou, Guiqin Qiu, Lingda Xu
{"title":"Asymptotic stability of the composite wave of rarefaction wave and contact wave to nonlinear viscoelasticity model with non-convex flux","authors":"Zhenhua Guo, Meichen Hou, Guiqin Qiu, Lingda Xu","doi":"arxiv-2409.10125","DOIUrl":"https://doi.org/arxiv-2409.10125","url":null,"abstract":"In this paper, we consider the wave propagations of viscoelastic materials,\u0000which has been derived by Taiping-Liu to approximate the viscoelastic dynamic\u0000system with fading memory (see [T.P.Liu(1988)cite{LiuTP}]) by the\u0000Chapman-Enskog expansion. By constructing a set of linear diffusion waves\u0000coupled with the high-order diffusion waves to achieve cancellations to\u0000approximate the viscous contact wave well and explicit expressions, the\u0000nonlinear stability of the composite wave is obtained by a continuum argument. It emphasis that, the stress function in our paper is a general non-convex\u0000function, which leads to several essential differences from strictly hyperbolic\u0000systems such as the Euler system. Our method is completely new and can be\u0000applied to more general systems and a new weighted Poincar'e type of\u0000inequality is established, which is more challenging compared to the convex\u0000case and this inequality plays an important role in studying systems with\u0000non-convex flux.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260674","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}
引用次数: 0
On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source 关于具有通量限制和逻辑源的抛物线-椭圆形趋化模型解的存在性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10121
Silvia Sastre-Gomez, J. Ignacio Tello
{"title":"On the existence of solutions for a parabolic-elliptic chemotaxis model with flux limitation and logistic source","authors":"Silvia Sastre-Gomez, J. Ignacio Tello","doi":"arxiv-2409.10121","DOIUrl":"https://doi.org/arxiv-2409.10121","url":null,"abstract":"In this paper we study the existence of solutions of a parabolic-elliptic\u0000system of partial differential equations describing the behaviour of a\u0000biological species $u$ and a chemical stimulus $v$ in a bounded and regular\u0000domain $Omega$ of $mathbb{R}^N$. The equation for $u$ is a parabolic equation\u0000with a nonlinear second order term of chemotaxis type with flux limitation as $\u0000-chi div (u |nabla psi|^{p-2} nabla v)$, for $p>1$. The chemical substance\u0000distribution $v$ satisfies the elliptic equation $-Delta v+v=u$. The evolution\u0000of $u$ is also determined by a logistic type growth term $mu u(1-u)$. The\u0000system is studied under homogeneous Neumann boundary conditions. The main\u0000result of the article is the existence of uniformly bounded solutions for\u0000$p<3/2$ and any $Nge 2$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260675","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}
引用次数: 0
Nonlinear nonlocal reaction-diffusion problem with local reaction 具有局部反应的非线性非局部反应-扩散问题
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10110
Aníbal Rodríguez-Bernal, Silvia Sastre-Gomez
{"title":"Nonlinear nonlocal reaction-diffusion problem with local reaction","authors":"Aníbal Rodríguez-Bernal, Silvia Sastre-Gomez","doi":"arxiv-2409.10110","DOIUrl":"https://doi.org/arxiv-2409.10110","url":null,"abstract":"In this paper we analyse the asymptotic behaviour of some nonlocal diffusion\u0000problems with local reaction term in general metric measure spaces. We find\u0000certain classes of nonlinear terms, including logistic type terms, for which\u0000solutions are globally defined with initial data in Lebesgue spaces. We prove\u0000solutions satisfy maximum and comparison principles and give sign conditions to\u0000ensure global asymptotic bounds for large times. We also prove that these\u0000problems possess extremal ordered equilibria and solutions, asymptotically,\u0000enter in between these equilibria. Finally we give conditions for a unique\u0000positive stationary solution that is globally asymptotically stable for\u0000nonnegative initial data. A detailed analysis is performed for logistic type\u0000nonlinearities. As the model we consider here lack of smoothing effect,\u0000important focus is payed along the whole paper on differences in the results\u0000with respect to problems with local diffusion, like the Laplacian operator.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260676","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}
引用次数: 0
Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits 非线性速度耦合对准：避免碰撞和微观到宏观平均场极限
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10501
Young-Pil Choi, Michał Fabisiak, Jan Peszek
{"title":"Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits","authors":"Young-Pil Choi, Michał Fabisiak, Jan Peszek","doi":"arxiv-2409.10501","DOIUrl":"https://doi.org/arxiv-2409.10501","url":null,"abstract":"We investigate the pressureless fractional Euler-alignment system with\u0000nonlinear velocity couplings, referred to as the $p$-Euler-alignment system.\u0000This model features a nonlinear velocity alignment force, interpreted as a\u0000density-weighted fractional $p$-Laplacian when the singularity parameter\u0000$alpha$ exceeds the spatial dimension $d$. Our primary goal is to establish\u0000the existence of solutions for strongly singular interactions ($alpha ge d$)\u0000and compactly supported initial conditions. We construct solutions as\u0000mean-field limits of empirical measures from a kinetic variant of the\u0000$p$-Euler-alignment system. Specifically, we show that a sequence of empirical\u0000measures converges to a finite Radon measure, whose local density and velocity\u0000satisfy the $p$-Euler-alignment system. Our results are the first to prove the\u0000existence of solutions to this system in multi-dimensional settings without\u0000significant initial data restrictions, covering both nonlinear ($p>2$) and\u0000linear ($p=2$) cases. Additionally, we establish global existence, uniqueness,\u0000and collision avoidance for the corresponding particle ODE system under\u0000non-collisional initial conditions, extending previous results for $1 le p le\u0000alpha + 2$. This analysis supports our mean-field limit argument and\u0000contributes to understanding alignment models with singular communication.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260671","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}
引用次数: 0
Hypersonic flow onto a large curved wedge and the dissipation of shock wave 流向大弯楔的超音速气流和冲击波的消散
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.10059
Dian Hu, Aifang Qu
{"title":"Hypersonic flow onto a large curved wedge and the dissipation of shock wave","authors":"Dian Hu, Aifang Qu","doi":"arxiv-2409.10059","DOIUrl":"https://doi.org/arxiv-2409.10059","url":null,"abstract":"For supersonic flow past an obstacle, experiments show that the flow field\u0000after shocks changes slightly for incoming flow with Mach number larger to 5,\u0000named hypersonic flow. Hypersonic similarity principle was first found by Qian\u0000for thin wedge by studying potential flow. In this paper, we explore the\u0000existence of smooth flow field after shock for hypersonic potential flow past a\u0000curved smooth wedge with neither smallness assumption on the height of the\u0000wedge nor that it is a BV perturbation of a line. The asymptotic behaviour of\u0000the shock is also analysed. We proved that for given Bernoulli constant of the\u0000incoming flow, there exists a sufficient large constant such that if the Mach\u0000number of the incoming flow is larger than it, then there exists a global shock\u0000wave attached to the tip of the wedge together with a smooth flow field between\u0000it and the wedge. The state of the flow after shock is in a neighbourhood of a\u0000curve that is determined by the wedge and the density of the incoming flow. If\u0000the slope of the wedge has a positive limit as $x$ goes to infinity, then the\u0000slope of the shock tends to that of the self-similar case that the same\u0000incoming flow past a straight wedge with slope of the limit. Specifically, we\u0000demonstrate that if the slope of the wedge is parallel to the incoming flow at\u0000infinity, then the strength of the shock will attenuate to zero at infinity.\u0000The restrictions on the surface of a wedge have been greatly relaxed compared\u0000to the previous works on supersonic flow past wedges. The method employed in\u0000this paper is characteristic decomposition, and the existence of the solution\u0000is obtained by finding an invariant domain of the solution based on geometry\u0000structures of the governing equations. The idea and the method used here may be\u0000helpful for other problems.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260678","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}
引用次数: 0
Domain characterization for Schrödinger operators with sub-quadratic singularity 具有次二次奇异性的薛定谔算子的域特性分析
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.09917
Giorgio Metafune, Motohiro Sobajima
{"title":"Domain characterization for Schrödinger operators with sub-quadratic singularity","authors":"Giorgio Metafune, Motohiro Sobajima","doi":"arxiv-2409.09917","DOIUrl":"https://doi.org/arxiv-2409.09917","url":null,"abstract":"We characterize the domain of the Schr\"odinger operators\u0000$S=-Delta+c|x|^{-alpha}$ in $L^p(mathbb{R}^N)$, with $0<alpha<2$ and\u0000$cinmathbb{R}$. When $alpha p< N$, the domain characterization is\u0000essentially known and can be proved using different tools, for instance kernel\u0000estimates and potentials in the Kato class or in the reverse H\"older class.\u0000However,the other cases seem not to be known, so far.In this paper, we give the\u0000explicit description of the domain of $S$ for all range of parameters\u0000$p,alpha$ and $c$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260680","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}
引用次数: 0
Fractional logarithmic Schrödinger equations on lattice graphs 网格图上的分数对数薛定谔方程
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-16 DOI: arxiv-2409.09976
Lidan Wang
{"title":"Fractional logarithmic Schrödinger equations on lattice graphs","authors":"Lidan Wang","doi":"arxiv-2409.09976","DOIUrl":"https://doi.org/arxiv-2409.09976","url":null,"abstract":"In this paper, we study the fractional logarithmic Schr\"{o}dinger equation\u0000$$ (-Delta)^{s} u+h(x) u=u log u^{2} $$ on lattice graphs $mathbb{Z}^d$,\u0000where $sin (0,1)$. If $h(x)$ is a bounded periodic potential, we prove the\u0000existence of ground state solution by mountain pass theorem and Lions lemma. If\u0000$h(x)$ is a coercive potential, we show the existence of ground state\u0000sign-changing solutions by the method of Nehari manifold.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260679","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}
引用次数: 0
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