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Discrete and Continuous Dynamical Systems

April 2022 , Volume 42 , Issue 4

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Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay
Ning Wang and Zhi-Cheng Wang
2022, 42(4): 1599-1646 doi: 10.3934/dcds.2021166 +[Abstract](671) +[HTML](207) +[PDF](2970.59KB)

This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.

Stability of optimal traffic plans in the irrigation problem
Maria Colombo, Antonio De Rosa, Andrea Marchese, Paul Pegon and Antoine Prouff
2022, 42(4): 1647-1667 doi: 10.3934/dcds.2021167 +[Abstract](611) +[HTML](198) +[PDF](482.04KB)

We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework.

Rigidity, weak mixing, and recurrence in abelian groups
Ethan M. Ackelsberg
2022, 42(4): 1669-1705 doi: 10.3934/dcds.2021168 +[Abstract](526) +[HTML](181) +[PDF](683.74KB)

The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about \begin{document}$ \mathbb{Z} $\end{document}-actions extend to this setting:

1. If \begin{document}$ (a_n) $\end{document} is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.

2. There exists a sequence \begin{document}$ (r_n) $\end{document} such that every translate is both a rigidity sequence and a set of recurrence.

The first of these results was shown for \begin{document}$ \mathbb{Z} $\end{document}-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in \begin{document}$ \mathbb{Z} $\end{document} by Griesmer [23]. While techniques for handling \begin{document}$ \mathbb{Z} $\end{document}-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.

As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in \begin{document}$ \mathbb{Z} $\end{document}, while others exhibit new phenomena.

On super-exponential divergence of periodic points for partially hyperbolic systems
Xiaolong Li and Katsutoshi Shinohara
2022, 42(4): 1707-1729 doi: 10.3934/dcds.2021169 +[Abstract](536) +[HTML](177) +[PDF](516.27KB)

We say that a diffeomorphism \begin{document}$ f $\end{document} is super-exponentially divergent if for every \begin{document}$ b>1 $\end{document} the lower limit of \begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document} diverges to infinity, where \begin{document}$ \mbox{Per}_n(f) $\end{document} is the set of all periodic points of \begin{document}$ f $\end{document} with period \begin{document}$ n $\end{document}. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any \begin{document}$ n $\end{document}-dimensional smooth closed manifold \begin{document}$ M $\end{document} where \begin{document}$ n\ge 3 $\end{document}, there exists a non-empty open subset \begin{document}$ \mathcal{O} $\end{document} of \begin{document}$ \mbox{Diff}^1(M) $\end{document} such that diffeomorphisms with super-exponentially divergent property form a dense subset of \begin{document}$ \mathcal{O} $\end{document} in the \begin{document}$ C^1 $\end{document}-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a \begin{document}$ C^r $\end{document}-residual subset of \begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document} is also shown.

Nonlinear nonlocal reaction-diffusion problem with local reaction
Aníbal Rodríguez-Bernal and Silvia Sastre-Gómez
2022, 42(4): 1731-1765 doi: 10.3934/dcds.2021170 +[Abstract](559) +[HTML](189) +[PDF](685.55KB)

In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.

Instability of the soliton for the focusing, mass-critical generalized KdV equation
Benjamin Dodson and Cristian Gavrus
2022, 42(4): 1767-1799 doi: 10.3934/dcds.2021171 +[Abstract](481) +[HTML](178) +[PDF](651.08KB)

In this paper we prove instability of the soliton for the focusing, mass-critical generalized KdV equation. We prove that the solution to the generalized KdV equation for any initial data with mass smaller than the mass of the soliton and close to the soliton in \begin{document}$ L^{2} $\end{document} norm must eventually move away from the soliton.

Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $
Hui Liu and Ling Zhang
2022, 42(4): 1801-1816 doi: 10.3934/dcds.2021172 +[Abstract](543) +[HTML](173) +[PDF](491.87KB)

In this paper, we prove that there exist at least two non-contractible closed Reeb orbits on every dynamically convex \begin{document}$ \mathbb{R}P^{2n-1} $\end{document}, and if all the closed Reeb orbits are non-degenerate, then there are at least \begin{document}$ n $\end{document} closed Reeb orbits, where \begin{document}$ n\geq2 $\end{document}, the main ingredient is that we generalize some theories developed by I. Ekeland and H. Hofer for closed characteristics on compact convex hypersurfaces in \begin{document}$ {{\bf R}}^{2n} $\end{document} to symmetric compact star-shaped hypersurfaces. In addition, we use Ekeland-Hofer theory to give a new proof of a theorem recently by M. Abreu and L. Macarini that every dynamically convex symmetric compact star-shaped hypersurface carries an elliptic symmetric closed characteristic.

Fujita type results for quasilinear parabolic inequalities with nonlocal terms
Roberta Filippucci and Marius Ghergu
2022, 42(4): 1817-1833 doi: 10.3934/dcds.2021173 +[Abstract](566) +[HTML](180) +[PDF](472.19KB)

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form

where \begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}, \begin{document}$ L_{\mathcal{A}} $\end{document} denotes a weakly \begin{document}$ m $\end{document}-coercive operator, which includes as prototype the \begin{document}$ m $\end{document}-Laplacian or the generalized mean curvature operator, \begin{document}$ p,\,q>0 $\end{document}, while \begin{document}$ K\ast u^p $\end{document} stands for the standard convolution operator between a weight \begin{document}$ K>0 $\end{document} satisfying suitable conditions at infinity and \begin{document}$ u^p $\end{document}. For problem \begin{document}$ (P^-) $\end{document} we obtain a Fujita type exponent while for \begin{document}$ (P^+) $\end{document} we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.

On $ L^1 $ estimates of solutions of compressible viscoelastic system
Yusuke Ishigaki
2022, 42(4): 1835-1853 doi: 10.3934/dcds.2021174 +[Abstract](459) +[HTML](169) +[PDF](457.04KB)

We consider the large time behavior of solutions of compressible viscoelastic system around a motionless state in a three-dimensional whole space. We show that if the initial data belongs to \begin{document}$ W^{2,1} $\end{document}, and is sufficiently small in \begin{document}$ H^4\cap L^1 $\end{document}, the solutions grow in time at the same rate as \begin{document}$ t^{\frac{1}{2}} $\end{document} in \begin{document}$ L^1 $\end{document} due to diffusion wave phenomena of the system caused by interaction between sound wave, viscous diffusion and elastic wave.

Sublacunary sets and interpolation sets for nilsequences
Anh N. Le
2022, 42(4): 1855-1871 doi: 10.3934/dcds.2021175 +[Abstract](487) +[HTML](155) +[PDF](464.13KB)

A set \begin{document}$ E \subset \mathbb{N} $\end{document} is an interpolation set for nilsequences if every bounded function on \begin{document}$ E $\end{document} can be extended to a nilsequence on \begin{document}$ \mathbb{N} $\end{document}. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here \begin{document}$ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $\end{document} with \begin{document}$ r_1 < r_2 < \ldots $\end{document} is sublacunary if \begin{document}$ \lim_{n \to \infty} (\log r_n)/n = 0 $\end{document}. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for \begin{document}$ 2 $\end{document}-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.

Random attractors for dissipative systems with rough noises
Luu Hoang Duc
2022, 42(4): 1873-1902 doi: 10.3934/dcds.2021176 +[Abstract](721) +[HTML](376) +[PDF](580.7KB)

We provide an analytic approach to study the asymptotic dynamics of rough differential equations, with the driving noises of Hölder continuity. Such systems can be solved with Lyons' theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths. Using the framework of random dynamical systems and random attractors, we prove the existence and upper semi-continuity of the global pullback attractor for dissipative systems perturbed by bounded noises. Moreover, if the unperturbed system is strictly dissipative then the random attractor is a singleton for sufficiently small noise intensity.

Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions
Corentin Le Bihan
2022, 42(4): 1903-1932 doi: 10.3934/dcds.2021177 +[Abstract](434) +[HTML](169) +[PDF](705.05KB)

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of \begin{document}$ N $\end{document} hard spheres of diameter \begin{document}$ \epsilon $\end{document} in a box \begin{document}$ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $\end{document}. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling \begin{document}$ N\epsilon^2 = 1 $\end{document}, \begin{document}$ \epsilon\rightarrow 0 $\end{document} to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem
Muhammad Hamid and Wei Wang
2022, 42(4): 1933-1948 doi: 10.3934/dcds.2021178 +[Abstract](460) +[HTML](172) +[PDF](463.62KB)

In this paper, we prove a symmetric property for the indices for symplectic paths in the enhanced common index jump theorem (cf. Theorem 3.5 in [6]). As an application of this property, we prove that on every compact Finsler manifold \begin{document}$ (M, \, F) $\end{document} with reversibility \begin{document}$ \lambda $\end{document} and flag curvature \begin{document}$ K $\end{document} satisfying \begin{document}$ \left(\frac{\lambda}{\lambda+1}\right)^2<K\le 1 $\end{document}, there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form \begin{document}$ e^{\sqrt {-1}\theta} $\end{document} with \begin{document}$ \frac{\theta}{\pi}\notin{\bf Q} $\end{document} provided the number of closed geodesics on \begin{document}$ M $\end{document} is finite.

Global propagation of singularities for discounted Hamilton-Jacobi equations
Cui Chen, Jiahui Hong and Kai Zhao
2022, 42(4): 1949-1970 doi: 10.3934/dcds.2021179 +[Abstract](464) +[HTML](175) +[PDF](516.38KB)

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation

with fixed constant \begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}. We reduce the problem for equation \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} propagate along locally Lipschitz singular characteristics \begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document} and time \begin{document}$ t $\end{document} can extend to \begin{document}$ +\infty $\end{document}. Essentially, we use \begin{document}$ \sigma $\end{document}-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of \begin{document}$ u $\end{document} and the complement of Aubry set using the basic idea from [9].

High and low perturbations of Choquard equations with critical reaction and variable growth
Youpei Zhang, Xianhua Tang and Vicenţiu D. Rădulescu
2022, 42(4): 1971-2003 doi: 10.3934/dcds.2021180 +[Abstract](574) +[HTML](170) +[PDF](624.17KB)

We are concerned with the existence of ground state solutions to the nonhomogeneous perturbed Choquard equation

where the exponent \begin{document}$ r(\cdot) $\end{document} is critical with respect to the Hardy-Littlewood-Sobolev inequality for variable exponents. We first consider the case where the perturbation \begin{document}$ g(\cdot ,\cdot) $\end{document} is subcritical and we distinguish between the superlinear and sublinear cases. In both situations we establish the existence of solutions and we prove the asymptotic behavior of low-energy solutions in the case of high perturbations. Next, we study the case where the nonlinearity \begin{document}$ g(\cdot ,\cdot) $\end{document} is critical. We prove the existence of solutions both for low and high perturbations and we establish asymptotic properties of low-energy solutions.

Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian
Sandro Zagatti
2022, 42(4): 2005-2025 doi: 10.3934/dcds.2021181 +[Abstract](505) +[HTML](174) +[PDF](479.13KB)

We study the minimum problem for functionals of the form

where the integrand \begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document} is not convex in the last variable. We provide an existence result assuming that the lower convex envelope \begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document} of \begin{document}$ f $\end{document} with respect to \begin{document}$ \xi $\end{document} is regular and enjoys a special dependence with respect to the i-th single components \begin{document}$ p_i, q_i, \xi_i $\end{document} of the vector variables \begin{document}$ p,q,\xi $\end{document}. More precisely, we assume that it is monotone in \begin{document}$ p_i $\end{document} and that it satisfies suitable affinity properties with respect to \begin{document}$ \xi_i $\end{document} on the set \begin{document}$ \{f> \overline{f}\} $\end{document} and with respect to \begin{document}$ q_i $\end{document} on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.

Crystalline flow starting from a general polygon
Mi-Ho Giga, Yoshikazu Giga, Ryo Kuroda and Yusuke Ochiai
2022, 42(4): 2027-2051 doi: 10.3934/dcds.2021182 +[Abstract](596) +[HTML](220) +[PDF](681.63KB)

This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification
Runlin Zhang
2022, 42(4): 2053-2071 doi: 10.3934/dcds.2021183 +[Abstract](552) +[HTML](157) +[PDF](539.47KB)

Let \begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document} be a semisimple linear algebraic group defined over rational numbers, \begin{document}$ \mathrm{K} $\end{document} be a maximal compact subgroup of its real points and \begin{document}$ \Gamma $\end{document} be an arithmetic lattice. One can associate a probability measure \begin{document}$ \mu_{ \mathrm{H}} $\end{document} on \begin{document}$ \Gamma \backslash \mathrm{G} $\end{document} for each subgroup \begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document} of \begin{document}$ \mathit{\boldsymbol{\mathrm{G}}} $\end{document} defined over \begin{document}$ \mathbb{Q} $\end{document} with no non-trivial rational characters. As G acts on \begin{document}$ \Gamma \backslash \mathrm{G} $\end{document} from the right, we can push forward this measure by elements from \begin{document}$ \mathrm{G} $\end{document}. By pushing down these measures to \begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document}, we call them homogeneous. It is a natural question to ask what are the possible weak-\begin{document}$ * $\end{document} limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of \begin{document}$ \Gamma \backslash \mathrm{G}/ \mathrm{K} $\end{document} for \begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document} generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general \begin{document}$ \mathit{\boldsymbol{\mathrm{H}}} $\end{document} with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on \begin{document}$ {\text{SL}}_n $\end{document} proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words.

2021 Impact Factor: 1.588
5 Year Impact Factor: 1.568
2021 CiteScore: 2.4




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