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Nondegeneracy and stability in the limit of a one-phase singular perturbation problem 一类单相奇异摄动问题的极限非退化性与稳定性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-02-26 DOI: 10.3934/dcds.2023089
Nikola Kamburov
{"title":"Nondegeneracy and stability in the limit of a one-phase singular perturbation problem","authors":"Nikola Kamburov","doi":"10.3934/dcds.2023089","DOIUrl":"https://doi.org/10.3934/dcds.2023089","url":null,"abstract":"We study solutions to a one-phase singular perturbation problem that arises in combustion theory and that formally approximates the classical one-phase free boundary problem. We introduce a natural density condition on the transition layers themselves that guarantees that the key nondegeneracy growth property of solutions is satisfied and preserved in the limit. We then apply our result to the problem of classifying global stable solutions of the underlying semilinear problem and we show that those have flat level sets in dimensions $nleq 4$, provided the density condition is fulfilled. The notion of stability that we use is the one with respect to inner domain deformations and in the process, we derive succinct new formulas for the first and second inner variations of general functionals of the form $I(v) = int |nabla v|^2 + mathcal{F}(v)$ that hold in a Riemannian manifold setting.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90176901","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}
引用次数: 0
Existence analysis for a reaction-diffusion Cahn–Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth 具有退化迁移率和奇异势的反应扩散cahn - hilliard型系统模拟生物膜生长的存在性分析
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-02-15 DOI: 10.3934/dcds.2023069
Christoph Helmer, A. Jungel
{"title":"Existence analysis for a reaction-diffusion Cahn–Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth","authors":"Christoph Helmer, A. Jungel","doi":"10.3934/dcds.2023069","DOIUrl":"https://doi.org/10.3934/dcds.2023069","url":null,"abstract":"The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation for the volume fraction of the biomass, considered in a bounded domain with no-flux boundary conditions. The main difficulties are coming from the degenerate diffusivity and mobility, the singular potential arising from a logarithmic free energy, and the nonlinear reaction rates. These issues are overcome by a truncation technique and a Browder-Minty trick to identify the weak limits of the reaction terms. The qualitative behavior of the solutions is illustrated by numerical experiments in one space dimension, using a BDF2 (second-order backward Differentiation Formula) finite-volume scheme.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"5 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83217267","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}
引用次数: 0
Reducibility for a linear wave equation with Sobolev smooth fast-driven potential 具有Sobolev光滑快速驱动势的线性波动方程的可约性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-19 DOI: 10.3934/dcds.2023047
L. Franzoi
{"title":"Reducibility for a linear wave equation with Sobolev smooth fast-driven potential","authors":"L. Franzoi","doi":"10.3934/dcds.2023047","DOIUrl":"https://doi.org/10.3934/dcds.2023047","url":null,"abstract":"We prove a reducibility result for a linear wave equation with a time quasi-periodic driving on the one dimensional torus. The driving is assumed to be fast oscillating, but not necessarily of small size. Provided that the external frequency vector is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, block-diagonal one. With the present paper we extend the previous work cite{FM19} to more general assumptions: we replace the analytic regularity in time with Sobolev one; the potential in the Schr\"odinger operator is a non-trivial smooth function instead of the constant one. The key tool to achieve the result is a localization property of each eigenfunction of the Schr\"odinger operator close to a subspace of exponentials, with a polynomial decay away from the latter.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"15 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89955508","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}
引用次数: 1
Existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces Besov-Morrey空间中分数阶半线性抛物方程解的存在性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-11 DOI: 10.3934/dcds.2023074
Erbol Zhanpeisov
{"title":"Existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces","authors":"Erbol Zhanpeisov","doi":"10.3934/dcds.2023074","DOIUrl":"https://doi.org/10.3934/dcds.2023074","url":null,"abstract":"In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"72 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79639739","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}
引用次数: 2
Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities Navier-Stokes和液晶不等式的前向奇异的抛物分形维数
3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023121
Gabriel S. Koch
{"title":"Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities","authors":"Gabriel S. Koch","doi":"10.3934/dcds.2023121","DOIUrl":"https://doi.org/10.3934/dcds.2023121","url":null,"abstract":"In 1985, V. Scheffer discussed partial regularity for what he called solutions to the 'Navier-Stokes inequality', which only satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system. One may extend this notion to a system introduced by F.-H. Lin and C. Liu in 1995 to model the flow of nematic liquid crystals, which include the Navier-Stokes system when the 'director field' $ d $ is taken to be zero. The model includes a further parabolic system which implies an a priori maximum principle for $ d $, which is lost when one considers the analogous 'inequality'.In 2018, Q. Liu proved a partial regularity result for solutions to the Lin-Liu model in terms of the 'parabolic fractal dimension' $ text{dim}_{ text{pf}} $, relying on the boundedness of $ d $ coming from the maximum principle. Q. Liu proves $ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac{95}{63}} $ for any compact $ mathcal{K} $, where $ Sigma_{-} $ is the set of space-time points near which the solution blows up forwards in time. For solutions to the corresponding 'inequality', we prove that, without any compensation for the lack of maximum principle, one has $ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac {55}{13}} $. We also provide a range of criteria, including as just one example the boundedness of $ d $, any one of which would furthermore imply that solutions to the inequality also satisfy $ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac{95}{63}} $.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"10 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135212657","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}
引用次数: 0
The connection between the dynamical properties of 3D systems and the image of the energy-Casimir mapping 三维系统的动力学特性与能量-卡西米尔映射图像之间的联系
3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023126
Mingxing Xu, Shaoyun Shi, Kaiyin Huang
{"title":"The connection between the dynamical properties of 3D systems and the image of the energy-Casimir mapping","authors":"Mingxing Xu, Shaoyun Shi, Kaiyin Huang","doi":"10.3934/dcds.2023126","DOIUrl":"https://doi.org/10.3934/dcds.2023126","url":null,"abstract":"We investigate the connection between the dynamical properties of a class of 3D systems and the geometric characteristics of the image of the energy-Casimir mapping. By examining the energy-Casimir mapping for such systems, we can explore the stability of the equilibrium states, the distribution of the periodic solutions, and the existence of homoclinic or heteroclinic orbits. We apply our findings to investigate the dynamic behavior of two specific equations, and provide a topological classification of the fibers of the energy-Casimir mapping for the two systems.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135213393","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}
引用次数: 0
Decay of correlations for some non-uniformly hyperbolic attractors 一些非一致双曲吸引子的相关衰减
3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023128
Sebastian Burgos
{"title":"Decay of correlations for some non-uniformly hyperbolic attractors","authors":"Sebastian Burgos","doi":"10.3934/dcds.2023128","DOIUrl":"https://doi.org/10.3934/dcds.2023128","url":null,"abstract":"We study the decay of correlations for certain dynamical systems with non-uniformly hyperbolic attractors, which natural invariant measure is the Sinai-Ruelle-Bowen (SRB) measure. The system $ g $ that we consider is produced by applying the slow-down procedure to a uniformly hyperbolic diffeomorphism $ f $ with an attractor. Under certain assumptions on the map $ f $ and the slow-down neighborhood, we show that the map $ g $ admits polynomial upper and lower bounds on correlations with respect to its SRB measure and the class of Hölder continuous observables. Our results apply to the Smale-Williams solenoid, as well as its sufficiently small perturbations.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"694 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135445025","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}
引用次数: 0
Quantitative Khintchine in simultaneous approximation 同时近似的定量钦钦机
3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023107
Shreyasi Datta
{"title":"Quantitative Khintchine in simultaneous approximation","authors":"Shreyasi Datta","doi":"10.3934/dcds.2023107","DOIUrl":"https://doi.org/10.3934/dcds.2023107","url":null,"abstract":"In a ground-breaking work [8], Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds, resolving a long-standing problem in the theory of Diophantine approximation. In this paper, we prove an effective version of their result.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135599898","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}
引用次数: 0
Blow-up for a fully fractional heat equation 放大成完全分数式热方程
3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023116
Raúl Ferreira, Arturo de Pablo
{"title":"Blow-up for a fully fractional heat equation","authors":"Raúl Ferreira, Arturo de Pablo","doi":"10.3934/dcds.2023116","DOIUrl":"https://doi.org/10.3934/dcds.2023116","url":null,"abstract":"We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation$ mathcal{M} u = u^p, qquad xinmathbb{R}^N, ;0<t<T $with $ p>0 $, where $ mathcal{M} $ is a nonlocal operator given by a space-time kernel $ M(x, t) = c_{N, sigma}t^{-frac N2-1-sigma}e^{-frac{|x|^2}{4t}} mathbb{1}_{{t>0}} $, $ 0<sigma<1 $. This operator coincides with the fractional power of the heat operator, $ mathcal{M} = (partial_t-Delta)^{sigma} $ defined through semigroup theory. We characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_* = 1+frac{2sigma}{N+2(1-sigma)} $. We also study the rate at which the blowing-up solutions below $ p_* $ tend to infinity, $ |u(cdot, t)|_inftysim (T-t)^{-fracsigma{p-1}} $.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136306274","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}
引用次数: 1
Global boundedness of a three-species predator-prey model with prey-taxis and competition 具有猎物趋向性和竞争的三种捕食者-猎物模型的全局有界性
IF 1.1 3区 数学
Discrete and Continuous Dynamical Systems Pub Date : 2023-01-01 DOI: 10.3934/dcds.2023061
Songzhi Li, Kaiqiang Wang
{"title":"Global boundedness of a three-species predator-prey model with prey-taxis and competition","authors":"Songzhi Li, Kaiqiang Wang","doi":"10.3934/dcds.2023061","DOIUrl":"https://doi.org/10.3934/dcds.2023061","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"62 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72491273","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}
引用次数: 2
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