# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2020 , Volume 40 , Issue 2

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2020, 40(2): 683-707 doi: 10.3934/dcds.2020057 +[Abstract](1188) +[HTML](129) +[PDF](482.1KB)
Abstract:

This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multi-classes. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. The limits of such systems as the number of particles tends to infinity are investigated. We establish the existence, uniqueness, and basic properties of solutions to the limiting McKean-Vlasov equations of these systems and then obtain the rate of convergence of the sequences of empirical measures associated with the systems to their limits in terms of the \begin{document}$p^{\text{th}}$\end{document} Monge-Wasserstein distance.

2020, 40(2): 709-724 doi: 10.3934/dcds.2020058 +[Abstract](1168) +[HTML](102) +[PDF](414.52KB)
Abstract:

We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on \begin{document}$n$\end{document} dimensional nontrapping asymptotically Euclidean manifolds, when \begin{document}$n = 3, 4$\end{document} as well as two dimensional Euclidean space. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.

2020, 40(2): 725-752 doi: 10.3934/dcds.2020059 +[Abstract](1111) +[HTML](98) +[PDF](464.63KB)
Abstract:

The dynamics is studied of an infinite collection of point particles placed in \begin{document}$\mathbb{R}^d$\end{document}, \begin{document}$d\geq 1$\end{document}. The particles perform random jumps with mutual repulsion accompanied by random merging of pairs of particles. The states of the collection are probability measures on the corresponding configuration space. The main result is the proof of the existence of the Markov evolution of states for a bounded time horizon if the initial state is a sub-Poissonian measure. The proof is based on representing sub-Poissonian measures \begin{document}$\mu$\end{document} by their correlation functions \begin{document}$k_\mu$\end{document} and is done in two steps: (a) constructing an evolution \begin{document}$k_{\mu_0} \to k_t$\end{document}; (b) proving that \begin{document}$k_t$\end{document} is the correlation function of a unique sub-Poissonian state \begin{document}$\mu_t$\end{document}.

2020, 40(2): 753-766 doi: 10.3934/dcds.2020060 +[Abstract](1128) +[HTML](93) +[PDF](322.3KB)
Abstract:

We consider the family of CIFSs of generalized complex continued fractions with a complex parameter space. This is a new interesting example to which we can apply a general theory of infinite CIFSs and analytic families of infinite CIFSs. We show that the Hausdorff dimension function of the family of the CIFSs of generalized complex continued fractions is continuous in the parameter space and is real-analytic and subharmonic in the interior of the parameter space. As a corollary of these results, we also show that the Hausdorff dimension function has a maximum point and the maximum point belongs to the boundary of the parameter space.

2020, 40(2): 767-780 doi: 10.3934/dcds.2020061 +[Abstract](1934) +[HTML](115) +[PDF](323.37KB)
Abstract:

We develop a thermodynamic formalism for a class of diffeomorphisms of a torus that are "almost-Anosov". In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium states for a collection of non-Hölder continuous geometric potentials over almost Anosov systems with an indifferent fixed point, as well as prove exponential decay of correlations and the central limit theorem for these equilibrium measures.

2020, 40(2): 781-815 doi: 10.3934/dcds.2020062 +[Abstract](1860) +[HTML](104) +[PDF](484.39KB)
Abstract:

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field:

where \begin{document}$\varepsilon>0$\end{document} is a small parameter, \begin{document}$a, b>0$\end{document} are constants, \begin{document}$s\in (\frac{3}{4}, 1)$\end{document}, \begin{document}$(-\Delta)^{s}_{A}$\end{document} is the fractional magnetic Laplacian, \begin{document}$A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$\end{document} is a smooth magnetic potential, \begin{document}$V:\mathbb{R}^{3}\rightarrow \mathbb{R}$\end{document} is a positive continuous electric potential satisfying local conditions and \begin{document}$f:\mathbb{R}\rightarrow \mathbb{R}$\end{document} is a \begin{document}$C^{1}$\end{document} subcritical nonlinearity. Applying penalization techniques, fractional Kato's type inequality and Ljusternik-Schnirelmann theory, we relate the number of nontrivial solutions with the topology of the set where the potential \begin{document}$V$\end{document} attains its minimum.

2020, 40(2): 817-845 doi: 10.3934/dcds.2020063 +[Abstract](1056) +[HTML](107) +[PDF](866.92KB)
Abstract:

We investigate the existence of a curve \begin{document}$q\mapsto u_{q}$\end{document}, with \begin{document}$q\in(0, 1)$\end{document}, of positive solutions for the problem

where \begin{document}$\Omega$\end{document} is a bounded and smooth domain of \begin{document}$\mathbb{R}^{N}$\end{document} and \begin{document}$a:\Omega\rightarrow\mathbb{R}$\end{document} is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of \begin{document}$u_{q}$\end{document} as \begin{document}$q\rightarrow0^{+}$\end{document} and \begin{document}$q\rightarrow1^{-}$\end{document}. We also show that in some cases \begin{document}$u_{q}$\end{document} is the ground state solution of \begin{document}$(P_{a, q})$\end{document}. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.

2020, 40(2): 847-881 doi: 10.3934/dcds.2020064 +[Abstract](1173) +[HTML](122) +[PDF](514.58KB)
Abstract:

In this paper, we investigate a sharp Moser-Trudinger inequality which involves the anisotropic Sobolev norm in unbounded domains. Under this anisotropic Sobolev norm, we establish the Lions type concentration-compactness alternative firstly. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality. In particular, we combine the low dimension case of \begin{document}$n = 2$\end{document} and the high dimension case of \begin{document}$n\geq 3$\end{document} to prove the existence of the extremal functions, which is different from the arguments of isotropic case, see [5,19].

2020, 40(2): 883-905 doi: 10.3934/dcds.2020065 +[Abstract](1185) +[HTML](86) +[PDF](403.98KB)
Abstract:

This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a \begin{document}$C^1$\end{document} diffeomorphism, utilizing the techniques in sub-additive thermodynamic formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which is exactly the sum of the dimensions of the restriction of the hyperbolic set to stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set vary continuously with respect to the dynamics.

2020, 40(2): 907-932 doi: 10.3934/dcds.2020066 +[Abstract](1016) +[HTML](86) +[PDF](430.5KB)
Abstract:

We investigate the diffusion-aggregation equations with degenerate diffusion \begin{document}$\Delta u^m$\end{document} and singular interaction kernel \begin{document}$\mathcal{K}_s = (-\Delta)^{-s}$\end{document} with \begin{document}$s\in(0,\frac{d}{2})$\end{document}. The equation is related to biological aggregation models. We analyze the regime where the diffusive force is stronger than the aggregation effect. In such regime, we show the existence and uniform boundedness of solutions in the case either \begin{document}$s>\frac{1}{2}$\end{document} or \begin{document}$m<2$\end{document}. Hölder regularity is obtained when \begin{document}$d\geq3, s>1/2$\end{document} and uniqueness is proved when \begin{document}$s\geq 1$\end{document}.

2020, 40(2): 933-982 doi: 10.3934/dcds.2020067 +[Abstract](1032) +[HTML](92) +[PDF](926.28KB)
Abstract:

We study global properties of positive radial solutions of \begin{document}$-\Delta u = u^p+M\left |{\nabla u}\right |^{\frac{2p}{p+1}}$\end{document} in \begin{document}$\mathbb R^N$\end{document} where \begin{document}$p>1$\end{document} and \begin{document}$M$\end{document} is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at \begin{document}$0$\end{document} according to the values of \begin{document}$M$\end{document} and \begin{document}$p$\end{document}.

2020, 40(2): 983-1012 doi: 10.3934/dcds.2020068 +[Abstract](1325) +[HTML](82) +[PDF](437.74KB)
Abstract:

We show that for Sturm-Liouville Systems on the half-line \begin{document}$[0, \infty)$\end{document}, the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at \begin{document}$x = 0$\end{document}. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of \begin{document}$L^2 ((0, \infty), \mathbb{C}^{n})$\end{document} solutions to the Sturm-Liouville System, and the case in which the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at \begin{document}$x = 0$\end{document}. In the former case, a formula of Hörmander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schrödinger equation on a star graph is linearized about a half-soliton solution.

2020, 40(2): 1013-1063 doi: 10.3934/dcds.2020069 +[Abstract](1190) +[HTML](109) +[PDF](609.72KB)
Abstract:

The existence of singular limit solutions are investigated by establishing a new Liouville type theorem for nonlinear elliptic system by using the Pohozaev type identity and the nonlinear domain decomposition method.

2020, 40(2): 1065-1073 doi: 10.3934/dcds.2020070 +[Abstract](1311) +[HTML](93) +[PDF](305.24KB)
Abstract:

We estimate the frequency of polynomial iterations which fall in a given multiplicative subgroup of a finite field of \begin{document}$p$\end{document} elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first \begin{document}$N$\end{document} elements in an orbit. We derive these from more general results about sequences of compositions on a fixed set of polynomials.

2020, 40(2): 1075-1105 doi: 10.3934/dcds.2020071 +[Abstract](975) +[HTML](108) +[PDF](1010.77KB)
Abstract:

We study the classification and evolution of bifurcation curves of positive solutions of the one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity given by

where \begin{document}$1/10\leq \varepsilon \leq1/5$\end{document}. It is interesting to find that the evolution of bifurcation curves is not completely identical with that for the one-dimensional perturbed Gelfand equations, even though it is the same for these two problems with zero Dirichlet boundary conditions. In fact, we prove that there exist a positive number \begin{document}$\varepsilon^{\ast}\,(\approx0.178)$\end{document} and three nonnegative numbers \begin{document}$c_{0}(\varepsilon)<c_{1}(\varepsilon)<c_{2}(\varepsilon)$\end{document} defined on \begin{document}$[1/10,1/5]$\end{document} with \begin{document}$c_{0} = 0$\end{document} if \begin{document}$1/10<\varepsilon \leq \varepsilon^{\ast}$\end{document} and \begin{document}$c_{0}>0$\end{document} if \begin{document}$\varepsilon^{\ast}<\varepsilon \leq1/5$\end{document}, such that, on the \begin{document}$(\lambda,\Vert u\Vert_{\infty})$\end{document}-plane, (ⅰ) when \begin{document}$0<c\leq c_{0}(\varepsilon)$\end{document} and \begin{document}$c\geq c_{2}(\varepsilon)$\end{document}, the bifurcation curve is strictly increasing; (ⅱ) when \begin{document}$c_{0}(\varepsilon)<c<c_{1}(\varepsilon)$\end{document}, the bifurcation curve is \begin{document}$S$\end{document}-shaped; (ⅲ) when \begin{document}$c_{1}(\varepsilon)\leq c<c_{2}(\varepsilon)$\end{document}, the bifurcation curve is \begin{document}$\subset$\end{document}-shaped.

2020, 40(2): 1107-1130 doi: 10.3934/dcds.2020072 +[Abstract](1113) +[HTML](83) +[PDF](373.5KB)
Abstract:

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line \begin{document}$\mathbb{R}^+$\end{document}, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

2020, 40(2): 1131-1157 doi: 10.3934/dcds.2020073 +[Abstract](1365) +[HTML](106) +[PDF](414.31KB)
Abstract:

Combining fixed point techniques with the method of lower-upper solutions we prove the existence of at least one weak solution for the following boundary value problem

where \begin{document}$\nu_{1}, \nu_{2}\in \mathbb{R}$\end{document}, \begin{document}$\Phi: \mathbb{R} \rightarrow \mathbb{R}$\end{document} is a strictly increasing homeomorphism extending the classical \begin{document}$p$\end{document}-Laplacian, \begin{document}$a$\end{document} is a nonnegative continuous function on \begin{document}$\mathbb{R} \times \mathbb{R}$\end{document} which can vanish on a set having zero Lebesgue measure and \begin{document}$f$\end{document} is a Carathéodory function on \begin{document}$\mathbb{R} \times \mathbb{R}^{2}$\end{document}.

2020, 40(2): 1159-1189 doi: 10.3934/dcds.2020074 +[Abstract](1093) +[HTML](104) +[PDF](468.38KB)
Abstract:

This paper is concerned with the following problem involving critical Sobolev exponent and polyharmonic operator:

where \begin{document}$B_1$\end{document} is the unit ball in \begin{document}$\mathbb{R}^{N}$\end{document}, \begin{document}$s_1$\end{document} and \begin{document}$\lambda$\end{document} are two positive parameters, \begin{document}$\varphi_1 > 0$\end{document} is the eigenfunction of \begin{document}$\left((-\Delta )^m, \mathcal{D}_0^{m,2}(B_1) \right)$\end{document} corresponding to the first eigenvalue \begin{document}$\lambda_1$\end{document} with \begin{document}$\hbox{ max }_{y \in B_1} \varphi_1(y) = 1$\end{document}, \begin{document}$u_+ = \hbox{ max }(u,0)$\end{document} and \begin{document}$m^* = \frac{2N}{N-2m}$\end{document}. By using the Lyapunov-Schmits reduction method, we prove that the number of solutions for \begin{document}$(P)$\end{document} is unbounded as the parameter \begin{document}$s_1$\end{document} tends to infinity, therefore proving the Lazer-McKenna conjecture for the higher order operator equation with critical growth.

2020, 40(2): 1191-1231 doi: 10.3934/dcds.2020075 +[Abstract](1631) +[HTML](109) +[PDF](615.62KB)
Abstract:

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space \begin{document}$L^2(0,1)^2$\end{document} according to the classical theory by Brézis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the \begin{document}$L^m$\end{document}-norms for all \begin{document}$m\in [1,+\infty]$\end{document} and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

2020, 40(2): 1233-1256 doi: 10.3934/dcds.2020076 +[Abstract](1122) +[HTML](116) +[PDF](453.93KB)
Abstract:

We prove some new Liouville-type theorems for stable radial solutions of

where \begin{document}$f$\end{document} is a smooth nonlinearity and \begin{document}$N \ge 2$\end{document}. Also, the sharpness of our results is discussed by means of some examples.

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