We study a coupled PDE system describing the dynamics of morphogen transport in epithelia, where the morphogens sense the spatial gradient of the logarithm of the signal following the empirically well-tested Webner–Fecher law. We prove that this fully parabolic system is globally well-posed and its unique solution is classical and uniformly bounded in time. Moreover, we find that regardless of the strength of the chemotactic motion and the size of the initial data, a linear degradation is strong enough to overcome the logarithmic singularity and destabilize the system globally and exponentially in time. Several numerical simulations are presented to illustrate and support the theoretical results.
{"title":"Global and exponential stabilization of morphogenesis models with logarithmic sensitivity and linear degradation","authors":"Lin Chen, Fanze Kong, Qi Wang","doi":"10.3934/dcds.2023115","DOIUrl":"https://doi.org/10.3934/dcds.2023115","url":null,"abstract":"We study a coupled PDE system describing the dynamics of morphogen transport in epithelia, where the morphogens sense the spatial gradient of the logarithm of the signal following the empirically well-tested Webner–Fecher law. We prove that this fully parabolic system is globally well-posed and its unique solution is classical and uniformly bounded in time. Moreover, we find that regardless of the strength of the chemotactic motion and the size of the initial data, a linear degradation is strong enough to overcome the logarithmic singularity and destabilize the system globally and exponentially in time. Several numerical simulations are presented to illustrate and support the theoretical results.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136366912","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}
Palmer's linearization theorem states that a hyperbolic linear system is topologically conjugated to its bounded perturbation. Recently, Huerta (DCDS 2020 [8]), Castañeda and Robledo (DCDS 2018 [3]) and Lin (NA 2007 [13]) generalized Palmer's theorem to the linearization with unbounded perturbation (continuous or discrete) by assuming that the linear part of the system is contractive or nonuniformly contractive. However, these previous works sacrifice the hyperbolicity of the linear part. Is it possible to study the linearization with unbounded perturbations in the hyperbolic case? In this paper, we improve the previous works [3,8,13] to the hyperbolic unbounded systems. For the contraction, each trajectory crosses its respective unit sphere exactly once. However, for the hyperbolic system, either the trajectory does not cross the unit sphere, or the trajectory cross it twice. Thus, the standard method used in the previous works for the contractive case is not valid for the hyperbolic case yet. We develop a method to overcome the difficulty based on two 'cylinders'. Furthermore, quantitative results for the parameters are provided.
帕尔默线性化定理指出一个双曲线性系统是拓扑共轭于它的有界摄动的。最近，Huerta (DCDS 2020 [8])， Castañeda和Robledo (DCDS 2018[3])和Lin (NA 2007[13])通过假设系统的线性部分是收缩或非均匀收缩，将Palmer定理推广到具有无界摄动(连续或离散)的线性化。然而，这些先前的作品牺牲了线性部分的双曲性。有没有可能研究双曲情况下无界扰动的线性化?本文将前人的研究成果[3,8,13]改进到双曲无界系统。对于收缩，每条轨迹正好穿过它各自的单位球一次。然而，对于双曲系统，要么轨迹没有穿过单位球，要么轨迹两次穿过单位球。因此，在前面的工作中使用的标准方法的收缩情况是无效的双曲情况。我们开发了一种基于两个“圆柱体”的方法来克服困难。此外，还给出了参数的定量结果。
{"title":"Linearization of a nonautonomous unbounded system with hyperbolic linear part: A spectral approach","authors":"Mengda Wu, Yonghui Xia","doi":"10.3934/dcds.2023112","DOIUrl":"https://doi.org/10.3934/dcds.2023112","url":null,"abstract":"Palmer's linearization theorem states that a hyperbolic linear system is topologically conjugated to its bounded perturbation. Recently, Huerta (DCDS 2020 [8]), Castañeda and Robledo (DCDS 2018 [3]) and Lin (NA 2007 [13]) generalized Palmer's theorem to the linearization with unbounded perturbation (continuous or discrete) by assuming that the linear part of the system is contractive or nonuniformly contractive. However, these previous works sacrifice the hyperbolicity of the linear part. Is it possible to study the linearization with unbounded perturbations in the hyperbolic case? In this paper, we improve the previous works [3,8,13] to the hyperbolic unbounded systems. For the contraction, each trajectory crosses its respective unit sphere exactly once. However, for the hyperbolic system, either the trajectory does not cross the unit sphere, or the trajectory cross it twice. Thus, the standard method used in the previous works for the contractive case is not valid for the hyperbolic case yet. We develop a method to overcome the difficulty based on two 'cylinders'. Furthermore, quantitative results for the parameters are provided.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136256735","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 work we consider the initial value problem for the generalized Benjamin equation begin{equation}label{Benj-IVP} begin{cases} partial_t u-lmathcal{H} partial_x^2u-partial_x^3u+u^ppartial_xu = 0, quad x,; tin mathbb{R};;;,; pgeq 1, u(x,0) = u_0(x), end{cases} end{equation} where $u=u(x,t)$ is a real valued function, $0
{"title":"Evolution of the radius of analyticity for the generalized Benjamin equation","authors":"Renata O. Figueira, M. Panthee","doi":"10.3934/dcds.2023039","DOIUrl":"https://doi.org/10.3934/dcds.2023039","url":null,"abstract":"In this work we consider the initial value problem for the generalized Benjamin equation begin{equation}label{Benj-IVP} begin{cases} partial_t u-lmathcal{H} partial_x^2u-partial_x^3u+u^ppartial_xu = 0, quad x,; tin mathbb{R};;;,; pgeq 1, u(x,0) = u_0(x), end{cases} end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83783217","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 stabilizability to trajectories of the Schl"ogl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls.
{"title":"Global stabilizability to trajectories for the Schlögl equation in a Sobolev norm","authors":"K. Kunisch, S. Rodrigues","doi":"10.3934/dcds.2023017","DOIUrl":"https://doi.org/10.3934/dcds.2023017","url":null,"abstract":"The stabilizability to trajectories of the Schl\"ogl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74343132","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 classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ mathbb{R}^4$: begin{equation}left{ begin{aligned}&-Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, quad xin mathbb{R}^4,&(-Delta)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, quad xin mathbb{R}^4, end{aligned} right. end{equation} where $ 0leq p_1<1$, $ p_2>0$, $ q_1>0$, $ q_2 geq 0$, $ u>0$ and satisfies $$ int_{mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx
{"title":"Classification of solutions for some mixed order elliptic system","authors":"Genggeng Huang, Yating Niu","doi":"10.3934/dcds.2023079","DOIUrl":"https://doi.org/10.3934/dcds.2023079","url":null,"abstract":"In this paper, we classify the solution of the following mixed-order conformally invariant system with coupled nonlinearity in $ mathbb{R}^4$: begin{equation}left{ begin{aligned}&-Delta u(x) = u^{p_1}(x) e^{q_1v(x)}, quad xin mathbb{R}^4,&(-Delta)^2 v(x) = u^{p_2}(x) e^{q_2v(x)}, quad xin mathbb{R}^4, end{aligned} right. end{equation} where $ 0leq p_1<1$, $ p_2>0$, $ q_1>0$, $ q_2 geq 0$, $ u>0$ and satisfies $$ int_{mathbb{R}^4} u^{p_1}(x) e^{q_1v(x)} dx<infty,quad int_{mathbb{R}^4} u^{p_2}(x) e^{q_2 v(x)} dx<infty.$$ Under additional assumptions (H1) or (H2), we study the asymptotic behavior of the solutions to the system and we establish the equivalent integral formula for the system. By using the method of moving spheres, we obtain the classification results of the solutions in the system.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82358092","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 note we study a new kinetic model of opinion dynamics. The model incorporates two forces -- alignment of opinions under all-to-all communication driving the system to a consensus, and Rayleigh type friction force that drives each `player' to its fixed conviction value. The balance between these forces creates a non-trivial limiting outcome. We establish existence of a global mono-opinion state, whereby any initial distribution of opinions for each conviction value aggregates to the Dirac measure concentrated on a single opinion. We identify several cases where such a state is unique and depends continuously on the initial distribution of convictions. Several regularity properties of the limiting distribution of opinions are presented.
{"title":"Continuous model of opinion dynamics with convictions","authors":"Vinh Nguyen, R. Shvydkoy","doi":"10.3934/dcds.2023076","DOIUrl":"https://doi.org/10.3934/dcds.2023076","url":null,"abstract":"In this note we study a new kinetic model of opinion dynamics. The model incorporates two forces -- alignment of opinions under all-to-all communication driving the system to a consensus, and Rayleigh type friction force that drives each `player' to its fixed conviction value. The balance between these forces creates a non-trivial limiting outcome. We establish existence of a global mono-opinion state, whereby any initial distribution of opinions for each conviction value aggregates to the Dirac measure concentrated on a single opinion. We identify several cases where such a state is unique and depends continuously on the initial distribution of convictions. Several regularity properties of the limiting distribution of opinions are presented.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91162837","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 propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.
{"title":"A space-time nonlocal traffic flow model: Relaxation representation and local limit","authors":"Q. Du, Kuang Huang, J. Scott, Wen Shen","doi":"10.3934/dcds.2023054","DOIUrl":"https://doi.org/10.3934/dcds.2023054","url":null,"abstract":"We propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79341606","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 show that every codimension one partially hyperbolic diffeomorphism must support on $mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.
{"title":"On codimension one partially hyperbolic diffeomorphisms","authors":"Xiang Zhang","doi":"10.3934/dcds.2023066","DOIUrl":"https://doi.org/10.3934/dcds.2023066","url":null,"abstract":"We show that every codimension one partially hyperbolic diffeomorphism must support on $mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78742126","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}
Y. Chitour, S'ebastien Fueyo, Guilherme Mazanti, M. Sigalotti
The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-Lambda_1,dots,t-Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q in [1, +infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $dmax{Lambda_1,dots,Lambda_N}$, where $d$ is the dimension of the state space.
本文研究有限维线性差分时滞方程的可控性，即给定时间$t$的状态可以用时间$t$的控制值和之前有限多个时间$t-Lambda_1,dots,t-Lambda_N$的状态值的线性组合来表示的动力学。基于yamamoto提出的一般无限维动力系统的实现理论，我们得到了在$L^q$空间，$q in [1, +infty)$中有限时间近似可控的频域充要条件。我们还提供了$L^1$精确可控性的一个必要条件，可以看作是$L^1$近似可控性判据的闭包。进一步，我们提供了近似和精确可控性的最小时间的显式上界，由$dmax{Lambda_1,dots,Lambda_N}$给出，其中$d$是状态空间的维数。
{"title":"Hautus–Yamamoto criteria for approximate and exact controllability of linear difference delay equations","authors":"Y. Chitour, S'ebastien Fueyo, Guilherme Mazanti, M. Sigalotti","doi":"10.3934/dcds.2023049","DOIUrl":"https://doi.org/10.3934/dcds.2023049","url":null,"abstract":"The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time $t$ and of the state evaluated at finitely many previous instants of time $t-Lambda_1,dots,t-Lambda_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q in [1, +infty)$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $dmax{Lambda_1,dots,Lambda_N}$, where $d$ is the dimension of the state space.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88155682","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 Lipschitz $mathbb{Z}-$periodic function $phi:mathbb{R}to mathbb{R}^2$ satisfied that $mathbb{R}^2setminus{phi(x):xinmathbb{R}}$ is not connected, an integer $bge 2$ and $lambdain (c/{b^{frac12}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=sumlimits_{n=0}^{infty}{{lambda}^nphi(b^nx)}$: the box dimension of its graph is equal to $3+2log_blambda$, where $c$ is a constant depending on $phi$.
{"title":"Box dimension of the graphs of the generalized Weierstrass-type functions","authors":"Haojie Ren","doi":"10.3934/dcds.2023068","DOIUrl":"https://doi.org/10.3934/dcds.2023068","url":null,"abstract":"For a Lipschitz $mathbb{Z}-$periodic function $phi:mathbb{R}to mathbb{R}^2$ satisfied that $mathbb{R}^2setminus{phi(x):xinmathbb{R}}$ is not connected, an integer $bge 2$ and $lambdain (c/{b^{frac12}},1)$, we prove the following for the generalized Weierstrass-type function $W(x)=sumlimits_{n=0}^{infty}{{lambda}^nphi(b^nx)}$: the box dimension of its graph is equal to $3+2log_blambda$, where $c$ is a constant depending on $phi$.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79385013","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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