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.
本文研究了燃烧理论中出现的一类单相奇异摄动问题的解，该问题的形式近似于经典的单相自由边界问题。我们在过渡层上引入一个自然密度条件，保证解的关键非简并生长性质满足并保持在极限内。然后，我们将我们的结果应用于对潜在半线性问题的全局稳定解进行分类的问题，并证明了在密度条件满足的情况下，这些问题在$nleq 4$维度上具有平坦的水平集。我们使用的稳定性概念是关于内域变形的，在这个过程中，我们为黎曼流形中一般泛函的第一次和第二次内变导出了简洁的新公式$I(v) = int |nabla v|^2 + mathcal{F}(v)$。
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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}} $.
1985年，V. Scheffer讨论了他所谓的“Navier-Stokes不等式”解的部分正则性，这些解只满足不可压缩条件以及局部和全局能量不等式和压力方程，这些方程可以从Navier-Stokes系统形式上推导出来。人们可以把这个概念扩展到f - h引入的系统。Lin和C. Liu在1995年建立了向列型液晶的流动模型，其中包括当“导向场”$ d $为零时的Navier-Stokes系统。该模型还包括一个进一步的抛物系统，该系统暗示了$ d $的先验极大值原理，当考虑类似的“不等式”时，该原理就丢失了。2018年，Q. Liu利用极大值原理中$ d $的有界性，证明了Lin-Liu模型在“抛物分形维数”$ text{dim}_{ text{pf}} $下解的部分正则性结果。Q. Liu证明了$ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac{95}{63}} $对于任意紧致$ mathcal{K} $，其中$ Sigma_{-} $是解在时间上向前爆炸的时空点的集合。对于相应的“不等式”的解，我们证明了，在不补偿极大值原理缺失的情况下，有$ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac {55}{13}} $。我们还提供了一系列准则，包括$ d $的有界性，其中任何一个都进一步暗示不等式的解也满足$ { text{dim}_{ text{pf}}(Sigma_{-} cap mathcal{K}) leq tfrac{95}{63}} $。
{"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}
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.
{"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}
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.
研究了一类具有非一致双曲吸引子的动力系统的相关性衰减，该系统的自然不变测度为Sinai-Ruelle-Bowen (SRB)测度。将慢化过程应用于具有吸引子的一致双曲微分同胚$ f $，得到了我们所考虑的系统$ g $。在映射$ f $和slow-down邻域的某些假设下，我们证明了映射$ g $与其SRB测度和Hölder连续可观测值类的相关性具有多项式上界和下界。我们的结果适用于small - williams螺线管，以及它的足够小的扰动。
{"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}
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.
{"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}
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, ;00 $, 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
{"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}
{"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}
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