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Discrete & Continuous Dynamical Systems

September 2021 , Volume 41 , Issue 9

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Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph
Shuichi Jimbo and Yoshihisa Morita
2021, 41(9): 4013-4039 doi: 10.3934/dcds.2021026 +[Abstract](350) +[HTML](162) +[PDF](696.14KB)

We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as \begin{document}$ t\to\pm\infty $\end{document} and classify the entire solutions according to their behaviors, where an entire solution is meant by a classical solution defined for all \begin{document}$ t\in(-\infty, \infty) $\end{document}. To this end, we give a condition under that the front propagation is blocked by the emergence of standing stationary solutions. The existence of an entire solution which propagates beyond the blocking is also shown.

On the dynamics of 3D electrified falling films
Jiao He and Rafael Granero-Belinchón
2021, 41(9): 4041-4064 doi: 10.3934/dcds.2021027 +[Abstract](254) +[HTML](131) +[PDF](722.63KB)

In this article, we consider a non-local variant of the Kuramoto-Sivashinsky equation in three dimensions (2D interface). Besides showing the global wellposedness of this equation we also obtain some qualitative properties of the solutions. In particular, we prove that the solutions become analytic in the spatial variable for positive time, the existence of a compact global attractor and an upper bound on the number of spatial oscillations of the solutions. We observe that such a bound is particularly interesting due to the chaotic behavior of the solutions.

Global generalized solutions to a chemotaxis model of capillary-sprout growth during tumor angiogenesis
Xueli Bai and Wenji Zhang
2021, 41(9): 4065-4083 doi: 10.3934/dcds.2021028 +[Abstract](259) +[HTML](134) +[PDF](447.07KB)

This paper considers a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis

under Neumann initial-boundary conditions in a smooth bounded domain. In the two-dimensional setting, introducing a generalized solution concept according to (Winkler, 2015 [35]) and constructing an appropriate regularized system, we prove the global existence of at least one such solution with suitably regular initial data by an approximation procedure. To overcome the difficulty in taking the limit to its regularized system, we establish some technical estimates related to several energy integrals with special structures like \begin{document}$ \int_0^T\int_{\Omega}{\frac{{{u_\varepsilon }v_\varepsilon^p}}{{1 +\varepsilon {u_\varepsilon }}}} $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ \int_0^T \int_\Omega {\frac{{{u_\varepsilon }}}{{1+\varepsilon {u_\varepsilon }}}\ln^{k}({u_\varepsilon } + 1)}dxdt $\end{document}, \begin{document}$ k\in(1,2) $\end{document}.

Best approximation of orbits in iterated function systems
Saisai Shi, Bo Tan and Qinglong Zhou
2021, 41(9): 4085-4104 doi: 10.3934/dcds.2021029 +[Abstract](300) +[HTML](133) +[PDF](398.14KB)

Let \begin{document}$ \Phi = \{\phi_{i}\colon i\in\Lambda\} $\end{document} be an iterated function system on a compact metric space \begin{document}$ (X,d) $\end{document}, where the index set \begin{document}$ \Lambda = \{1, 2, \ldots,l\} $\end{document} with \begin{document}$ l \ge2 $\end{document}, or \begin{document}$ \Lambda = \{1,2,\ldots\} $\end{document}. We denote by \begin{document}$ J $\end{document} the attractor of \begin{document}$ \Phi $\end{document}, and by \begin{document}$ D $\end{document} the subset of points possessing multiple codings. For any \begin{document}$ x\in J\backslash D, $\end{document} there is a unique integer sequence \begin{document}$ \{\omega_{n}(x)\}_{n\geq 1}\subset \Lambda^{\mathbb{N}} $\end{document}, called the digit sequence of \begin{document}$ x, $\end{document} such that

In this case we write \begin{document}$ x = [\omega_{1}(x),\omega_{2}(x),\ldots]. $\end{document} For \begin{document}$ x, y\in J\backslash D, $\end{document} we define the shortest distance function \begin{document}$ M_{n}(x,y) $\end{document} as

which counts the run length of the longest same block among the first \begin{document}$ n $\end{document} digits of \begin{document}$ (x,y). $\end{document}

In this paper, we are concerned with the asymptotic behaviour of \begin{document}$ M_{n}(x,y) $\end{document} as \begin{document}$ n $\end{document} tends to \begin{document}$ \infty. $\end{document} We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance function. As applications, we study the exceptional sets in several concrete systems such as continued fractions system, Lüroth system, \begin{document}$ N $\end{document}-ary system, and triadic Cantor system.

On multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension $ n $
Yanheng Ding, Xiaojing Dong and Qi Guo
2021, 41(9): 4105-4123 doi: 10.3934/dcds.2021030 +[Abstract](238) +[HTML](128) +[PDF](422.37KB)

In this paper, we study multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension \begin{document}$ n $\end{document}:

where \begin{document}$ n\geq 2 $\end{document}, \begin{document}$ \hbar>0 $\end{document} is a small parameter, \begin{document}$ a>0 $\end{document} is a constant, and \begin{document}$ f $\end{document} describes the self-interaction which is either subcritical: \begin{document}$ W(x)|u|^{p-2} $\end{document}, or critical: \begin{document}$ W_{1}(x)|u|^{p-2}+W_{2}(x)|u|^{2^*-2} $\end{document}, with \begin{document}$ p\in (2,2^*), 2^* = \frac{2n}{n-1} $\end{document}. The number of solutions obtained depending on the ratio of \begin{document}$ \min V $\end{document} and \begin{document}$ \liminf\limits_{|x|\rightarrow \infty} V(x) $\end{document}, as well as \begin{document}$ \max W $\end{document} and \begin{document}$ \limsup\limits_{|x|\rightarrow \infty} W(x) $\end{document} for the subcritical case and \begin{document}$ \max W_{j} $\end{document} and \begin{document}$ \limsup\limits_{|x|\rightarrow \infty} W_{j}(x), j = 1,2, $\end{document} for the critical case.

Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $
Claudianor O. Alves and Tahir Boudjeriou
2021, 41(9): 4125-4144 doi: 10.3934/dcds.2021031 +[Abstract](213) +[HTML](115) +[PDF](436.45KB)

In this paper we study the local and global existence of solutions for a class of heat equation in whole \begin{document}$ \mathbb{R}^{N} $\end{document} where the nonlinearity has a critical growth for \begin{document}$ N \geq 2 $\end{document}. In order to prove the global existence, we will use the potential well theory combined with the Nehari manifold, and also with the Pohozaev manifold that is a novelty for this type of problem. Moreover, the blow-up phenomena of local solutions is investigated by combining the subdifferential approach with the concavity method.

Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching
Dan Li and Hui Wan
2021, 41(9): 4145-4183 doi: 10.3934/dcds.2021032 +[Abstract](295) +[HTML](137) +[PDF](514.95KB)

This work investigates the dynamics of competitive Kolmogorov systems formulated in a semi-Markov regime-switching framework. The conditional holding time of each environmental regime is allowed to follow arbitrary probability distribution on the nonnegative half-line in the sense of approximations. Sharp sufficient conditions of the coexistence and competitive exclusion of species are established, and in the case of species coexistence, the convergence rate of the transition probability to the unique stationary measure is estimated. In weaker conditions, these results extend the existing results to the semi-Markov regime-switching environment. Particularly, the method of proving the exponential convergence of the transition probability to the invariant measure for the population models formulated as random differential equations driven by a semi-Markov process is proposed.

Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations
Mostafa Fazly and Yuan Li
2021, 41(9): 4185-4206 doi: 10.3934/dcds.2021033 +[Abstract](239) +[HTML](120) +[PDF](460.98KB)

We study the quasilinear elliptic equation

where the operator \begin{document}$ Q $\end{document}, known as the Finsler-Laplacian (or anisotropic Laplacian) operator, is defined by

Here \begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}} $\end{document} and \begin{document}$ F: \mathbb{R}^{N}\rightarrow[0, +\infty) $\end{document} is a convex function of \begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document} that satisfies certain assumptions. For a bounded domain \begin{document}$ \Omega $\end{document} and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed \begin{document}$ N-10 $\end{document}. For the case of entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions \begin{document}$ N<10 $\end{document} and \begin{document}$ 2<N<10 $\end{document}, respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions \begin{document}$ N = 2 $\end{document}. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities.

Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data
Shaoming Guo, Xianfeng Ren and Baoxiang Wang
2021, 41(9): 4207-4253 doi: 10.3934/dcds.2021034 +[Abstract](284) +[HTML](157) +[PDF](589.25KB)

Considering the Cauchy problem of the derivative NLS

we will show its local well-posedness in modulation spaces \begin{document}$ M^{1/2}_{2,q}(\mathbb{R}) $\end{document} \begin{document}$ (4{\leqslant} q<\infty) $\end{document}. It is well-known that \begin{document}$ H^{1/2} $\end{document} is a critical Sobolev space of the derivative NLS. Noticing that \begin{document}$ H^{1/2} \subset M^{1/2}_{2,q} \subset B^{1/q}_{2,q} $\end{document} \begin{document}$ (q{\geqslant} 2) $\end{document} are sharp inclusions, our result contains a class of functions in \begin{document}$ L^2\setminus H^{1/2} $\end{document}.

Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues
Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim and Doo Seok Lee
2021, 41(9): 4255-4281 doi: 10.3934/dcds.2021035 +[Abstract](278) +[HTML](130) +[PDF](1952.82KB)

In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers \begin{document}$ {\lambda}_0 $\end{document} and \begin{document}$ {\lambda}_1 $\end{document} for the control parameter \begin{document}$ {\lambda} $\end{document} in the equation. Motivated by [9], we assume that \begin{document}$ {\lambda}_0< {\lambda}_1 $\end{document} and the linearized operator at the trivial solution has multiple critical eigenvalues \begin{document}$ \beta_N^+ $\end{document} and \begin{document}$ \beta_{N+1}^+ $\end{document}. Then, we show that as \begin{document}$ {\lambda} $\end{document} passes through \begin{document}$ {\lambda}_0 $\end{document}, the trivial solution bifurcates to an \begin{document}$ S^1 $\end{document}-attractor \begin{document}$ {\mathcal A}_N $\end{document}. We verify that \begin{document}$ {\mathcal A}_N $\end{document} consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.

Singular double-phase systems with variable growth for the Baouendi-Grushin operator
Anouar Bahrouni and Vicenţiu D. Rădulescu
2021, 41(9): 4283-4296 doi: 10.3934/dcds.2021036 +[Abstract](678) +[HTML](127) +[PDF](349.33KB)

In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [6] to the case of lack of compactness corresponding to the whole Euclidean space. After establishing a related compactness property, we establish the existence of solutions for the Baouendi-Grushin singular system.

Supercritical elliptic problems involving a Cordes like operator
Craig Cowan
2021, 41(9): 4297-4318 doi: 10.3934/dcds.2021037 +[Abstract](213) +[HTML](84) +[PDF](464.46KB)

In this work we obtain positive bounded solutions of various perturbations of

where \begin{document}$ B_1 $\end{document} is the unit ball in \begin{document}$ {{\mathbb{R}}}^N $\end{document} where \begin{document}$ N \ge 3 $\end{document}, \begin{document}$ \gamma>0 $\end{document} and \begin{document}$ 1<p<p_{N, \gamma} $\end{document} where

Note for \begin{document}$ \gamma>0 $\end{document} this allows for supercritical range of \begin{document}$ p $\end{document}.

Permutations with restricted movement
Dor Elimelech
2021, 41(9): 4319-4349 doi: 10.3934/dcds.2021038 +[Abstract](205) +[HTML](91) +[PDF](676.34KB)

A restricted permutation of a locally finite directed graph \begin{document}$ G = (V, E) $\end{document} is a vertex permutation \begin{document}$ \pi: V\to V $\end{document} for which \begin{document}$ (v, \pi(v))\in E $\end{document}, for any vertex \begin{document}$ v\in V $\end{document}. The set of such permutations, denoted by \begin{document}$ \Omega(G) $\end{document}, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18] of restricted \begin{document}$ {\mathbb Z}^d $\end{document} permutations, in which \begin{document}$ \Omega(G) $\end{document} is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted \begin{document}$ {\mathbb Z}^d $\end{document}-permutations. We discuss the global and local admissibility of patterns, in the context of restricted \begin{document}$ {\mathbb Z}^d $\end{document}-permutations. Finally, we review the related models of injective and surjective restricted functions.

PDEs on graphs for semi-supervised learning applied to first-person activity recognition in body-worn video
Hao Li, Honglin Chen, Matt Haberland, Andrea L. Bertozzi and P. Jeffrey Brantingham
2021, 41(9): 4351-4373 doi: 10.3934/dcds.2021039 +[Abstract](200) +[HTML](126) +[PDF](2975.15KB)

This paper showcases the use of PDE-based graph methods for modern machine learning applications. We consider a case study of body-worn video classification because of the large volume of data and the lack of training data due to sensitivity of the information. Many modern artificial intelligence methods are turning to deep learning which typically requires a lot of training data to be effective. They can also suffer from issues of trust because the heavy use of training data can inadvertently provide information about details of the training images and could compromise privacy. Our alternate approach is a physics-based machine learning that uses classical ideas like optical flow for video analysis paired with linear mixture models such as non-negative matrix factorization along with PDE-based graph classification methods that parallel geometric equations from PDE such as motion by mean curvature. The upshot is a methodology that can work well on video with modest amounts of training data and that can also be used to compress the information about the video scene so that no personal information is contained in the compressed data, making it possible to provide a larger group of people access to these compressed data without compromising privacy. The compressed data retains information about the wearer of the camera while discarding information about people, objects, and places in the scene.

Flows with the weak two-sided limit shadowing property
Jihoon Lee and Ngocthach Nguyen
2021, 41(9): 4375-4395 doi: 10.3934/dcds.2021040 +[Abstract](240) +[HTML](83) +[PDF](389.52KB)

In this paper we study the weak two-sided limit shadowing for flows on a compact metric space which is different with the usual shadowing, two-sided limit shadowing and L-shadowing, and characterize the weak two-sided limit shadowing flows from the pointwise and measurable viewpoints. Moreover, we prove that if a flow \begin{document}$ \phi $\end{document} has the weak two-sided limit shadowing property on its chain recurrent \begin{document}$ CR(\phi) $\end{document} then the set \begin{document}$ CR(\phi) $\end{document} is decomposed by a finite number of closed invariant sets on which \begin{document}$ \phi $\end{document} is topologically transitive and has the two-sided limit shadowing property.

Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system
Qiao Liu
2021, 41(9): 4397-4419 doi: 10.3934/dcds.2021041 +[Abstract](185) +[HTML](93) +[PDF](523.14KB)

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an \begin{document}$ \varepsilon $\end{document}-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for \begin{document}$ \sigma\in [0,1] $\end{document}, there exists a \begin{document}$ \varepsilon>0 $\end{document} such that if \begin{document}$ (u,d,P) $\end{document} is a suitable weak solution in \begin{document}$ Q_{r}(z_{0}) $\end{document} with \begin{document}$ 0<r\leq 1 $\end{document} and \begin{document}$ z_{0} = (x_{0},t_{0}) $\end{document}, and satisfies

then \begin{document}$ (u, d) $\end{document} is regular at \begin{document}$ z_{0} $\end{document}. Here, \begin{document}$ H^{-\sigma}(B_{r}(x)) $\end{document} is the dual space of \begin{document}$ H^{\sigma}_{0}(B_{r}(x)) $\end{document}, the space of functions \begin{document}$ f $\end{document} in the homogeneous Sobolev space \begin{document}$ \dot{H}^{\sigma}(\mathbb{R}^{3}) $\end{document} such that \begin{document}$ \operatorname{supp} f\subset \overline{B_{r}(x)} $\end{document}. Inspired by this \begin{document}$ \varepsilon $\end{document}-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from \begin{document}$ \frac{95}{63} (\approx 1.50794) $\end{document} given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to \begin{document}$ \frac{835}{613} (\approx 1.36215) $\end{document}.

Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L tensor coefficient under relaxed ellipticity condition
Mirela Kohr, Sergey E. Mikhailov and Wolfgang L. Wendland
2021, 41(9): 4421-4460 doi: 10.3934/dcds.2021042 +[Abstract](235) +[HTML](110) +[PDF](631.89KB)

The first aim of this paper is to show well-posedness of Dirichlet and transmission problems in bounded and exterior Lipschitz domains in \begin{document}$ {\mathbb R}^n $\end{document}, \begin{document}$ n\geq 3 $\end{document}, for the anisotropic Stokes system with \begin{document}$ L_{\infty } $\end{document} viscosity tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices with zero matrix trace, with data from standard and weighted Sobolev spaces. To this end we reduce the linear problems to equivalent mixed variational formulations and show that the variational problems are well-posed. Then we use the Leray-Schauder fixed point theorem and establish the existence of a weak solution for nonlinear Dirichlet and transmission problems for the anisotropic Navier-Stokes system in bounded Lipschitz domains in \begin{document}$ {\mathbb R}^3 $\end{document}, with general (including large) data in Sobolev spaces. For exterior domains in \begin{document}$ {\mathbb R}^3 $\end{document}, the analysis of the nonlinear Dirichlet and transmission problems in weighted Sobolev spaces relies on the existence result for the Dirichlet problem for the anisotropic Navier-Stokes system in a family of bounded Lipschitz domains. The obtained estimates for pressure in \begin{document}$ {\mathbb R}^3 $\end{document} look new also for the classical isotropic case.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations
Minh-Phuong Tran and Thanh-Nhan Nguyen
2021, 41(9): 4461-4476 doi: 10.3934/dcds.2021043 +[Abstract](346) +[HTML](97) +[PDF](412.32KB)

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document}-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1<p \le \frac{3n-2}{2n-1} $\end{document}, where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document}. Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

Centralizers of partially hyperbolic diffeomorphisms in dimension 3
Thomas Barthelmé and Andrey Gogolev
2021, 41(9): 4477-4484 doi: 10.3934/dcds.2021044 +[Abstract](178) +[HTML](93) +[PDF](289.54KB)

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu [10] who recently classified the centralizer for perturbations of time-\begin{document}$ 1 $\end{document} maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in [5,6].

Equilibrium states for non-uniformly hyperbolic systems: Statistical properties and analyticity
Suzete Maria Afonso, Vanessa Ramos and Jaqueline Siqueira
2021, 41(9): 4485-4513 doi: 10.3934/dcds.2021045 +[Abstract](225) +[HTML](84) +[PDF](624.19KB)

We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.

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