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Discrete and Continuous Dynamical Systems

October 2022 , Volume 42 , Issue 10

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Positive solutions to a nonlinear fractional equation with an external source term
Qi Li, Kefan Pan and Shuangjie Peng
2022, 42(10): 4669-4706 doi: 10.3934/dcds.2022068 +[Abstract](362) +[HTML](98) +[PDF](465.22KB)

This paper deals with the following nonlinear fractional equation with an external source term

where \begin{document}$ N>2s $\end{document}, \begin{document}$ 0<s<1 $\end{document}, \begin{document}$ 1<p<2_{\ast}(s)-1 $\end{document}, \begin{document}$ 2_{\ast}(s) = \frac{2N}{N-2s} $\end{document}, \begin{document}$ K(x) $\end{document} is a continuous function and \begin{document}$ f\in L^{2}({\Bbb R}^{N})\cap L^{\infty}({\Bbb R}^{N}) $\end{document}. Using a Lyapunov-Schmidt reduction scheme, we prove that the equation admits \begin{document}$ k $\end{document}-peak solutions for any integer \begin{document}$ k>0 $\end{document} if \begin{document}$ f $\end{document} is small and \begin{document}$ K(x) $\end{document} satisfies some additional assumptions at infinity. The main difficulty is to improve the estimate of the remainder obtained in the reduction process.

Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $ \mathit{\boldsymbol{\mathbb{R}^N}} $
Hiroshi Matsuzawa and Mitsunori Nara
2022, 42(10): 4707-4740 doi: 10.3934/dcds.2022069 +[Abstract](374) +[HTML](172) +[PDF](485.11KB)

We consider the Cauchy problem for an anisotropic reaction diffusion equation with a multi-stable nonlinearity on \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\ge 2 $\end{document} and investigate the large time behavior of the solution. This problem with a bistable nonlinearity has been investigated by Matano, Mori and Nara [Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), pp. 585-626]. They showed that under the suitable condition on the initial function the solution develops a spreading front whose position is closely approximated by the expanding Wulff shape for all large times. In this paper we will extend their results to the problem with a multi-stable type nonlinearity, that is, the case where the nonlinearity can be decomposed to \begin{document}$ K $\end{document} number of bistable nonlinearities and show that under certain conditions on the nonlinearity and the initial function the solution develops \begin{document}$ K $\end{document} number of spreading fronts whose positions are closely approximated by the expanding Wulff shapes with different expanding speeds. In other words, for any direction the solution on the ray of the direction looks like stacked traveling waves, that is, on the ray the solution approaches the so called propagating terrace. The key step for extension to multi-stable case is to construct \begin{document}$ K $\end{document} number of upper solutions and lower solutions all at once.

Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality
Huaiyu Zhou and Jingbo Dou
2022, 42(10): 4741-4760 doi: 10.3934/dcds.2022070 +[Abstract](499) +[HTML](76) +[PDF](450.02KB)

In this paper, we classify the positive solutions to the following integral system

where \begin{document}$ N\ge1, p_j>1, 0<h_{ij}<N $\end{document} for all \begin{document}$ i, j\in\{1, 2, \cdots, k\} $\end{document}. Up to a positive constant multiplier, this system is the Euler-Lagrangian equations associated to the multilinear fractional integral inequality established by Beckner. Employing the method of moving spheres, we give the explicit form of positive solutions to the above system with \begin{document}$ p_j = \frac{\sum_{1\leq i<j\leq k} \ \ h_{ij}-(k-3)N}{(k-1)N-\sum_{1\leq i<j\leq k} \ \ h_{ij}} $\end{document} satisfying

and show the nonexistence of positive solutions for \begin{document}$ p_j>1 $\end{document} with

On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth
Shengbing Deng and Xingliang Tian
2022, 42(10): 4761-4786 doi: 10.3934/dcds.2022071 +[Abstract](433) +[HTML](74) +[PDF](457.04KB)

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} containing the origin with smooth boundary, \begin{document}$ \beta\in [0, 2) $\end{document}, \begin{document}$ m $\end{document} is a Kirchhoff type function, \begin{document}$ \|u_j\|^2 = \int_\Omega|\nabla u_j|^2dx $\end{document}, \begin{document}$ f_i $\end{document} behaves like \begin{document}$ e^{\alpha_0 s^2} $\end{document} when \begin{document}$ |s|\rightarrow \infty $\end{document} for some \begin{document}$ \alpha_0>0 $\end{document}, and there is \begin{document}$ C^1 $\end{document} function \begin{document}$ F: \Omega\times\mathbb{R}^k\to \mathbb{R} $\end{document} such that \begin{document}$ \left(\frac{\partial F}{\partial u_1}, \ldots, \frac{\partial F}{\partial u_k}\right) = \left(f_1, \ldots, f_k\right) $\end{document}, \begin{document}$ h_i\in \left(\big(H^1_0(\Omega)\big)^*, \|\cdot\|_*\right) $\end{document}. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when \begin{document}$ \varepsilon>0 $\end{document} is small.

Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows
Vladislav Kibkalo and Tomoo Yokoyama
2022, 42(10): 4787-4822 doi: 10.3934/dcds.2022072 +[Abstract](310) +[HTML](78) +[PDF](764.44KB)

It is known that \begin{document}$ C^r $\end{document} Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any \begin{document}$ r \in \mathbb{Z}_{\geq 1} $\end{document}. In particular, \begin{document}$ C^r $\end{document} Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of \begin{document}$ C^r $\end{document} gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows and instantaneous non-Morse gradient flows. To illustrate the generic transitions (e.g. bifurcations of singular points, transitions via heteroclinic separatrices), we characterize and list all generic non-Morse gradient flows. To construct such characterizations, we characterize isolated singular points of gradient flows on surfaces. In fact, such a singular point is a non-trivial finitely sectored singular point without elliptic sectors. Moreover, considering Morse-Smale flows as "generic gradient flows with limit cycles", we characterize and list all generic non-Morse-Smale flows.

Statistical properties of type D dispersing billiards
Margaret Brown and Péter Nándori
2022, 42(10): 4823-4851 doi: 10.3934/dcds.2022073 +[Abstract](242) +[HTML](59) +[PDF](480.92KB)

We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian [9]. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties.

Existence of invariant curves for degenerate almost periodic reversible mappings
Peng Huang
2022, 42(10): 4853-4886 doi: 10.3934/dcds.2022074 +[Abstract](274) +[HTML](64) +[PDF](445.93KB)

In this paper we are concerned with the existence of invariant curves for almost periodic reversible mappings with higher order degeneracy of the twist condition. In the proof we use a new variant of the KAM theory, containing an artificial parameter \begin{document}$ q, 0<q<1 $\end{document}, which makes the steps of the KAM iteration infinitely small in the speed of function \begin{document}$ q^n \varepsilon, $\end{document} rather than super exponential function.

On the asymptotic behavior of solutions for the self-dual Maxwell-Chern-Simons $ O(3) $ Sigma model
Zhi-You Chen, Chung-Yang Wang and Yu-Jen Huang
2022, 42(10): 4887-4903 doi: 10.3934/dcds.2022077 +[Abstract](315) +[HTML](138) +[PDF](371.81KB)

In this paper, we consider the nonlinear equations arising from the self-dual Maxwell-Chern-Simons gauged \begin{document}$ O(3) $\end{document} sigma model on (2+1)-dimensional Minkowski space \begin{document}$ {\bf R^{2,1}} $\end{document} with the metric \begin{document}$ {\mathrm {diag}}(1,-1,-1) $\end{document}. We establish the asymptotic behavior of multivortex solutions corresponding to their flux and find the range of the flux for non-topological solutions. Moreover, we prove the radial symmetry property under certain conditions in one vortex point case.

Zero-diffusion limit for aggregation equations over bounded domains
Razvan C. Fetecau, Hui Huang, Daniel Messenger and Weiran Sun
2022, 42(10): 4905-4936 doi: 10.3934/dcds.2022078 +[Abstract](355) +[HTML](56) +[PDF](1662.66KB)

We investigate the zero-diffusion limit for both continuous and discrete aggregation-diffusion models over convex and bounded domains. Our approach relies on a coupling method connecting PDEs with their underlying SDEs. Compared with existing work, our result relaxes the regularity assumptions on the interaction and external potentials and improves the convergence rate (in terms of the diffusion coefficient). The particular rate we derive is shown to be consistent with numerical computations.

On multi-solitons for coupled Lowest Landau Level equations
Laurent Thomann
2022, 42(10): 4937-4964 doi: 10.3934/dcds.2022081 +[Abstract](250) +[HTML](59) +[PDF](413.01KB)

We consider a coupled system of nonlinear Lowest Landau Level equations. We first show the existence of multi-solitons with an exponentially localised error term in space, and then we prove a uniqueness result. We also show a long time stability result of the sum of traveling waves having all the same speed, under the condition that they are localised far away enough from each other. Finally, we observe that these multi-solitons provide examples of dynamics for the linear Schrödinger equation with harmonic potential perturbed by a time-dependent potential.

Iterative roots of type $ \mathcal {T}_2 $
Liu Liu, Justyna Jarczyk, Witold Jarczyk and Weinian Zhang
2022, 42(10): 4965-4990 doi: 10.3934/dcds.2022082 +[Abstract](303) +[HTML](71) +[PDF](526.8KB)

This paper aims to an open problem on iterative roots of PM functions, a class of non-monotonic functions. The open problem asks: Does a PM function of nonmonotonicity height \begin{document}$ \ge 2 $\end{document} have a continuous iterative root of order \begin{document}$ n $\end{document} being less than or equal to the number of forts? It was proved that iterative roots of order \begin{document}$ n $\end{document} being equal to the number of forts (if exist) can be classified into two types: mostly increasing ones (type \begin{document}$ \mathcal {T}_1 $\end{document}) and mostly decreasing ones (type \begin{document}$ \mathcal {T}_2 $\end{document}) and all roots of type \begin{document}$ \mathcal {T}_1 $\end{document} are found, but the remaining type \begin{document}$ \mathcal {T}_2 $\end{document} is left for more complicated construction. In this paper full description of type \begin{document}$ \mathcal {T}_2 $\end{document} roots is given.

Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem
Weiwei Ao, Mengdie Fu and Chao Liu
2022, 42(10): 4991-5015 doi: 10.3934/dcds.2022083 +[Abstract](231) +[HTML](56) +[PDF](400.41KB)

Given a smooth bounded domain \begin{document}$ \Omega\subset{{\mathbb R}}^2 $\end{document}, we consider the following Ambrosetti-Prodi problem with Neumann boundary:

where \begin{document}$ p>2 $\end{document}, \begin{document}$ \sigma>0 $\end{document} is a large parameter and \begin{document}$ \nu $\end{document} denotes the outward normal of \begin{document}$ \partial \Omega $\end{document}. We constructed a new class of solutions comprised of a large number of spikes concentrated on a segment of the boundary containing a local minimum point of the mean curvature function and having the same mean curvature at the endpoints. A similar boundary-concentrating phenomenon was obtained for the Lin-Ni-Takagi problem by Ao et al. [3].

An optimization problem in heat conduction with volume constraint and double obstacles
Xiaoliang Li and Cong Wang
2022, 42(10): 5017-5036 doi: 10.3934/dcds.2022084 +[Abstract](257) +[HTML](59) +[PDF](387.76KB)

We consider the optimization problem of minimizing \begin{document}$ \int_{\mathbb{R}^n}|\nabla u|^2\,{\mathrm{d}}x $\end{document} with double obstacles \begin{document}$ \phi\leq u\leq\psi $\end{document} a.e. in \begin{document}$ D $\end{document} and a constraint on the volume of \begin{document}$ \{u>0\}\setminus\overline{D} $\end{document}, where \begin{document}$ D\subset\mathbb{R}^n $\end{document} is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is \begin{document}$ C^{1,1} $\end{document} locally in \begin{document}$ D $\end{document} and Lipschitz continuous in \begin{document}$ \mathbb{R}^n $\end{document} and that the free boundary \begin{document}$ \partial\{u>0\}\setminus\overline{D} $\end{document} is smooth. Moreover, when the boundary of \begin{document}$ D $\end{document} has a plane portion, we show that the minimizer is \begin{document}$ C^{1,\frac{1}{2}} $\end{document} up to the plane portion.

Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian
Martin Mayer and Cheikh Birahim Ndiaye
2022, 42(10): 5037-5062 doi: 10.3934/dcds.2022085 +[Abstract](230) +[HTML](58) +[PDF](414.59KB)

We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $\end{document} and fractional parameter in \begin{document}$ (\frac{1}{2}, 1) $\end{document} corresponding to a global case left by previous works.

Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem with large exponents
Habibi Sadaf and Futoshi Takahashi
2022, 42(10): 5063-5086 doi: 10.3934/dcds.2022086 +[Abstract](261) +[HTML](61) +[PDF](416.44KB)

In this paper we are concerned with the least energy solutions to the Lane-Emden problem driven by an anisotropic operator, so-called the Finsler \begin{document}$ N $\end{document}-Laplacian, on a bounded domain in \begin{document}$ {\mathbb{R}}^N $\end{document}. We prove several asymptotic formulae as the nonlinear exponent gets large.

Oriented and orbital shadowing for vector fields
Ming Li and Xingzhong Liu
2022, 42(10): 5087-5104 doi: 10.3934/dcds.2022087 +[Abstract](281) +[HTML](66) +[PDF](358.05KB)

In this paper, we prove that the \begin{document}$ C^1 $\end{document}-interior of the set of vector fields having the oriented shadowing property coincides with the \begin{document}$ C^1 $\end{document}-interior of the set of vector fields having the orbital shadowing property.

Ground states for a system of nonlinear Schrödinger equations with singular potentials
Peng Chen and Xianhua Tang
2022, 42(10): 5105-5136 doi: 10.3934/dcds.2022088 +[Abstract](285) +[HTML](70) +[PDF](424.27KB)

In this paper, we consider the existence and asymptotic behavior of ground state solutions for a class of Hamiltonian elliptic system with Hardy potential. The resulting problem engages three major difficulties: one is that the associated functional is strongly indefinite, the second difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is different from the usual global super-quadratic condition. The third difficulty is singular potential, which does not belong to the Kato's class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity of potential and the dropping of classical super-quadratic assumption on the nonlinearity. Our approach is based on non-Nehari method which developed recently, we establish some new existence results of ground state solutions of Nehari-Pankov type under some mild conditions, and analyze asymptotical behavior of ground state solutions.

Closed relations with non-zero entropy that generate no periodic points
Iztok Banič, Goran Erceg and Judy Kennedy
2022, 42(10): 5137-5166 doi: 10.3934/dcds.2022089 +[Abstract](237) +[HTML](55) +[PDF](459.89KB)

The paper is motivated by E. Akin's book about dynamical systems and closed relations [1], and by J. Kennedy's and G. Erceg's recent paper about the entropy of closed relations on closed intervals [9].

In present paper, we introduce the entropy of a closed relation \begin{document}$ G $\end{document} on any compact metric space \begin{document}$ X $\end{document} and show its basic properties. We also introduce when such a relation \begin{document}$ G $\end{document} generates a periodic point or finitely generates a Cantor set. Then we show that periodic points, finitely generated Cantor sets, Mahavier products and the entropy of closed relations are preserved by topological conjugations. Among other things, this generalizes the well-known results about the topological conjugacy of continuous mappings. Finally, we prove a theorem, giving sufficient conditions for a closed relation \begin{document}$ G $\end{document} on \begin{document}$ [0,1] $\end{document} to have a non-zero entropy. Then we present various examples of closed relations \begin{document}$ G $\end{document} on \begin{document}$ [0,1] $\end{document} such that (1) the entropy of \begin{document}$ G $\end{document} is non-zero, (2) no periodic point or exactly one periodic point is generated by \begin{document}$ G $\end{document}, and (3) no Cantor set is finitely generated by \begin{document}$ G $\end{document}.

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




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