# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete and Continuous Dynamical Systems

January 2022 , Volume 42 , Issue 1

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2022, 42(1): 1-19 doi: 10.3934/dcds.2021106 +[Abstract](1532) +[HTML](312) +[PDF](413.39KB)
Abstract:

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.

2022, 42(1): 21-71 doi: 10.3934/dcds.2021107 +[Abstract](1053) +[HTML](268) +[PDF](697.04KB)
Abstract:

We study the second-order boundary value problem

where \begin{document}$a_{\lambda,\mu}$\end{document} is a step-wise indefinite weight function, precisely \begin{document}$a_{\lambda,\mu}\equiv\lambda$\end{document} in \begin{document}$[0,\sigma]\cup[1-\sigma,1]$\end{document} and \begin{document}$a_{\lambda,\mu}\equiv-\mu$\end{document} in \begin{document}$(\sigma,1-\sigma)$\end{document}, for some \begin{document}$\sigma\in\left(0,\frac{1}{2}\right)$\end{document}, with \begin{document}$\lambda$\end{document} and \begin{document}$\mu$\end{document} positive real parameters. We investigate the topological structure of the set of positive solutions which lie in \begin{document}$(0,1)$\end{document} as \begin{document}$\lambda$\end{document} and \begin{document}$\mu$\end{document} vary. Depending on \begin{document}$\lambda$\end{document} and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter \begin{document}$\mu$\end{document}. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the \begin{document}$(\lambda,\mu)$\end{document}-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.

2022, 42(1): 73-108 doi: 10.3934/dcds.2021108 +[Abstract](980) +[HTML](225) +[PDF](528.52KB)
Abstract:

Consider the second order nonautonomous lattice systemswith singular perturbations

and the first order nonautonomous lattice systems

Under certain conditions, there are pullback attractors \begin{document}$\{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}}$\end{document} and \begin{document}$\{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}}$\end{document} for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors \begin{document}$\mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2}$\end{document}, \begin{document}$t\in \mathbb{R}$\end{document}, with respect to the coefficient \begin{document}$\epsilon$\end{document} of second derivative term under Hausdorff semidistance. First, we studythe relationship between \begin{document}$\mathcal{A}_{\epsilon }(t)$\end{document} and \begin{document}$\mathcal{A}(t)$\end{document} when \begin{document}$\epsilon \rightarrow 0^{+}$\end{document}. We construct a family of compact sets \begin{document}$\mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2}$\end{document}, \begin{document}$t\in \mathbb{R}$\end{document} such that \begin{document}$\mathcal{A}(t)$\end{document} is naturally embedded into \begin{document}$\mathcal{A}_{0}(t)$\end{document} as the firstcomponent, and prove that \begin{document}$\mathcal{A}_{\epsilon }(t)$\end{document} can enter anyneighborhood of \begin{document}$\mathcal{A}_{0}(t)$\end{document} when \begin{document}$\epsilon$\end{document} is small enough. Thenfor \begin{document}$\epsilon _{0}>0$\end{document}, we prove that \begin{document}$\mathcal{A}_{\epsilon }(t)$\end{document} can enterany neighborhood of \begin{document}$\mathcal{A}_{\epsilon _{0}}(t)$\end{document} when \begin{document}$\epsilon\rightarrow \epsilon _{0}$\end{document}. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).

2022, 42(1): 109-135 doi: 10.3934/dcds.2021109 +[Abstract](755) +[HTML](125) +[PDF](433.07KB)
Abstract:

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using a Lyapunov functional and duality arguments, we establish the existence of component wise non-negative global solutions.

2022, 42(1): 137-162 doi: 10.3934/dcds.2021110 +[Abstract](879) +[HTML](227) +[PDF](339.19KB)
Abstract:

We consider the following semilinear problem with a gradient term in the nonlinearity

where \begin{document}$\lambda,p,q>0$\end{document} and \begin{document}$\Omega$\end{document} be a bounded, smooth domain in \begin{document}${\mathbb R}^N$\end{document}. We prove that when \begin{document}$\Omega$\end{document} is a unit ball and \begin{document}$p = 1$\end{document} for \begin{document}$q\in (0,q^*(N))$\end{document} with \begin{document}$q^*(N)\in (1,2)$\end{document}, we have infinitely many radial solutions for \begin{document}$2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1$\end{document} and \begin{document}$\lambda = \tilde \lambda$\end{document}. On the other hand, for \begin{document}$N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1$\end{document} there exists a unique radial solution for \begin{document}$0<\lambda<\tilde \lambda$\end{document}.

2022, 42(1): 163-187 doi: 10.3934/dcds.2021111 +[Abstract](882) +[HTML](263) +[PDF](467.36KB)
Abstract:

In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities

where \begin{document}$\Omega$\end{document} is a smooth bounded domain of \begin{document}$\mathbb R^n$\end{document}, \begin{document}$n\geq 1$\end{document}, \begin{document}$s\in (0,1)$\end{document}, \begin{document}$\mu>0$\end{document} is a real parameter, \begin{document}$\beta <{n/(n-s)}$\end{document} and \begin{document}$q\in (0,1)$\end{document}.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.

2022, 42(1): 189-210 doi: 10.3934/dcds.2021112 +[Abstract](722) +[HTML](215) +[PDF](493.79KB)
Abstract:

We extend recent orbit counts for finitely generated semigroups acting on \begin{document}$\mathbb{P}^N$\end{document} to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.

2022, 42(1): 211-237 doi: 10.3934/dcds.2021113 +[Abstract](730) +[HTML](193) +[PDF](372.71KB)
Abstract:

This work establishes the relaxation limit from the bipolar Euler-Poisson system to the bipolar drift-diffusion system, for data so that the latter has a smooth solution. A relative energy identity is developed for the bipolar fluid models, and it is used to show that a dissipative weak solution of the bipolar Euler-Poisson system converges in the high-friction regime to a strong and bounded away from vacuum solution of the bipolar drift-diffusion system.

2022, 42(1): 239-259 doi: 10.3934/dcds.2021114 +[Abstract](720) +[HTML](252) +[PDF](417.38KB)
Abstract:

We study bifurcations in finite-parameter families of vector fields on \begin{document}$S^2$\end{document}. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable \begin{document}$3$\end{document}-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of \begin{document}$(2D+1)$\end{document}-parameter families such that the topological classification of these families has at least \begin{document}$D$\end{document} numerical invariants and used those examples to construct families with functional invariants of topological classification.

In this paper, we construct locally generic \begin{document}$4$\end{document}-parameter families with any prescribed number of numerical invariants and use them to construct \begin{document}$5$\end{document}-parameter families with functional invariants. We also describe a locally generic class of \begin{document}$3$\end{document}-parameter families with a tail of an infinite number sequence as an invariant of topological classification.

2022, 42(1): 261-283 doi: 10.3934/dcds.2021115 +[Abstract](755) +[HTML](201) +[PDF](504.06KB)
Abstract:

This paper is devoted to an anisotropic curvature flow of the form \begin{document}$V = A(\mathbf{n})H + B(\mathbf{n})$\end{document} in a band domain \begin{document}$\Omega : = [-1,1]\times {\mathbb{R}}$\end{document}, where \begin{document}$\mathbf{n}$\end{document}, \begin{document}$V$\end{document} and \begin{document}$H$\end{document} denote respectively the unit normal vector, normal velocity and curvature of a graphic curve \begin{document}$\Gamma_t$\end{document}. We require that the curve \begin{document}$\Gamma_t$\end{document} contacts \begin{document}$\partial \Omega$\end{document} with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the uniform interior gradient estimates for the solutions by using the zero number argument. Furthermore, when \begin{document}$t\to \infty$\end{document}, we show that \begin{document}$\Gamma_t$\end{document} converges to a traveling wave with cup-shaped profile and infinite boundary slopes in the \begin{document}$C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}})$\end{document}-topology.

2022, 42(1): 285-299 doi: 10.3934/dcds.2021116 +[Abstract](806) +[HTML](185) +[PDF](362.63KB)
Abstract:

Let \begin{document}$G$\end{document} be a countable monoid and let \begin{document}$A$\end{document} be an Artinian group (resp. an Artinian module). Let \begin{document}$\Sigma \subset A^G$\end{document} be a closed subshift which is also a subgroup (resp. a submodule) of \begin{document}$A^G$\end{document}. Suppose that \begin{document}$\Gamma$\end{document} is a finitely generated monoid consisting of pairwise commuting cellular automata \begin{document}$\Sigma \to \Sigma$\end{document} that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the natural action of \begin{document}$\Gamma$\end{document} on \begin{document}$\Sigma$\end{document} satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissible group subshifts.

2022, 42(1): 301-317 doi: 10.3934/dcds.2021117 +[Abstract](980) +[HTML](218) +[PDF](417.48KB)
Abstract:

Let \begin{document}$\Omega\subset\mathbb{R}^n \; (n\geq 2)$\end{document} be a bounded domain with continuous boundary \begin{document}$\partial\Omega$\end{document}. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to \begin{document}$\Omega$\end{document} with \begin{document}$0<s<1$\end{document}. Denoting by \begin{document}$\lambda_{k}$\end{document} the \begin{document}$k^{th}$\end{document} Dirichlet eigenvalue of \begin{document}$(-\triangle)^{s}|_{\Omega}$\end{document}, we establish the explicit upper bounds of the ratio \begin{document}$\frac{\lambda_{k+1}}{\lambda_{1}}$\end{document}, which have polynomially growth in \begin{document}$k$\end{document} with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function \begin{document}$R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma}$\end{document} with \begin{document}$\sigma\geq 1$\end{document} and the trace of the Dirichlet heat kernel of fractional Laplacian.

2022, 42(1): 319-326 doi: 10.3934/dcds.2021118 +[Abstract](762) +[HTML](186) +[PDF](304.0KB)
Abstract:

In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component \begin{document}$\mathcal{H}$\end{document} still lies on \begin{document}$\partial \mathcal{H}$\end{document}. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.

2022, 42(1): 327-352 doi: 10.3934/dcds.2021119 +[Abstract](689) +[HTML](192) +[PDF](450.76KB)
Abstract:

The goal of this paper is the study of the integrability of the geodesic flow on \begin{document}$k$\end{document}-step nilpotent Lie groups, k = 2, 3, when equipped with a left-invariant metric. Liouville integrability is proved in low dimensions. Moreover, it is shown that complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields. This holds for dimension \begin{document}$m\leq 5$\end{document}. The situation in dimension six is similar in most cases. Several algebraic relations on the Lie algebra of first integrals are explicitly written. Also invariant first integrals are analyzed and several involution conditions are shown.

2022, 42(1): 353-368 doi: 10.3934/dcds.2021120 +[Abstract](686) +[HTML](200) +[PDF](396.69KB)
Abstract:

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.

2022, 42(1): 369-401 doi: 10.3934/dcds.2021121 +[Abstract](887) +[HTML](176) +[PDF](517.56KB)
Abstract:

Let \begin{document}$H: = -\Delta+V$\end{document} be a nonnegative Schrödinger operator on \begin{document}$L^2({\bf R}^N)$\end{document}, where \begin{document}$N\ge 2$\end{document} and \begin{document}$V$\end{document} is a radially symmetric inverse square potential. Let \begin{document}$\|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})}$\end{document} be the operator norm of \begin{document}$\nabla^\alpha e^{-tH}$\end{document} from the Lorentz space \begin{document}$L^{p, \sigma}({\bf R}^N)$\end{document} to \begin{document}$L^{q, \theta}({\bf R}^N)$\end{document}, where \begin{document}$\alpha\in\{0, 1, 2, \dots\}$\end{document}. We establish both of upper and lower decay estimates of \begin{document}$\|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})}$\end{document} and study sharp decay estimates of \begin{document}$\|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})}$\end{document}. Furthermore, we characterize the Laplace operator \begin{document}$-\Delta$\end{document} from the view point of the decay of \begin{document}$\|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})}$\end{document}.

2022, 42(1): 403-423 doi: 10.3934/dcds.2021122 +[Abstract](860) +[HTML](191) +[PDF](553.1KB)
Abstract:

In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.

2022, 42(1): 425-462 doi: 10.3934/dcds.2021123 +[Abstract](884) +[HTML](184) +[PDF](425.63KB)
Abstract:

In this work, we will introduce a general framework to derive the thermodynamics of a fluid mechanical system, which guarantees the consistence between the energetic variational approaches with the laws of thermodynamics. In particular, we will focus on the coupling between the thermal and mechanical forces. We follow the framework for a classical gas with ideal gas equilibrium and present the existences of weak solutions to this thermodynamic system coupled with the Brinkman-type equation to govern the velocity field.

2022, 42(1): 463-479 doi: 10.3934/dcds.2021124 +[Abstract](726) +[HTML](180) +[PDF](433.82KB)
Abstract:

The exact null-controllability problem in the class of smooth controls with applications to interconnected systems was considered in [23] for the case of bounded input operators appearing in systems with distributed controls. The current paper constitutes an extension of the [23] for the case of unbounded input operators (with more emphasis on the controllability of interconnected systems). The proofs of the results of [23] for the case of bounded input operators are adopted for the case of unbounded input operators.

2022, 42(1): 481-504 doi: 10.3934/dcds.2021125 +[Abstract](825) +[HTML](185) +[PDF](425.74KB)
Abstract:

In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely \begin{document}$-\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u)$\end{document} in \begin{document}$\Omega$\end{document} with \begin{document}$u = 0$\end{document} on \begin{document}$\partial\Omega$\end{document}, where \begin{document}$\Omega$\end{document} is a smooth bounded domain in \begin{document}$\mathbb{R}^{2}$\end{document}. We replace the singular function \begin{document}$\log(u)$\end{document} by a function \begin{document}$g_\epsilon(u)$\end{document} which pointwisely converges to -\begin{document}$\log(u)$\end{document} as \begin{document}$\epsilon \rightarrow 0$\end{document}. When the parameter \begin{document}$\lambda>0$\end{document} is small enough, the corresponding energy functional to the perturbed equation \begin{document}$-\Delta u + g_\epsilon(u) = \lambda f(u)$\end{document} has a critical point \begin{document}$u_\epsilon$\end{document} in \begin{document}$H_0^1(\Omega)$\end{document}, which converges to a nontrivial nonnegative solution of the original problem as \begin{document}$\epsilon \rightarrow 0$\end{document}.

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