# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

All Issues

## Discrete and Continuous Dynamical Systems

Open Access Articles

2022, 42(4): 2027-2051 doi: 10.3934/dcds.2021182 +[Abstract](548) +[HTML](208) +[PDF](681.63KB)
Abstract:

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.

2022, 42(3): 1435-1463 doi: 10.3934/dcds.2021159 +[Abstract](517) +[HTML](164) +[PDF](527.93KB)
Abstract:

In the paper we study what sets can be obtained as \begin{document}$\alpha$\end{document}-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those \begin{document}$\alpha$\end{document}-limit sets are \begin{document}$\omega$\end{document}-limit sets and for all but finitely many points \begin{document}$x$\end{document}, we can obtain every \begin{document}$\omega$\end{document}-limits set as the \begin{document}$\alpha$\end{document}-limit set of a backward trajectory starting in \begin{document}$x$\end{document}. For zero entropy maps, every \begin{document}$\alpha$\end{document}-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

2021, 41(11): 5057-5085 doi: 10.3934/dcds.2021068 +[Abstract](1136) +[HTML](345) +[PDF](454.96KB)
Abstract:

In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.

2021, 41(8): 3985-4012 doi: 10.3934/dcds.2021025 +[Abstract](1014) +[HTML](281) +[PDF](1731.32KB)
Abstract:

In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish \begin{document}$\Gamma$\end{document}-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.

2021, 41(9): 4013-4039 doi: 10.3934/dcds.2021026 +[Abstract](1295) +[HTML](481) +[PDF](696.14KB)
Abstract:

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.

2021, 41(6): 2971-2992 doi: 10.3934/dcds.2020393 +[Abstract](1003) +[HTML](323) +[PDF](440.58KB)
Abstract:

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document} for \begin{document}$s \ge \frac14$\end{document} and \begin{document}$2\leq p < \infty$\end{document}. For \begin{document}$s < \frac 14$\end{document}, we show that the solution map for mKdV is not locally uniformly continuous in \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document}. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in \begin{document}$M^{2,p}_{s}( \mathbb{R})$\end{document} for \begin{document}$s \ge \frac14$\end{document} and \begin{document}$2\leq p < \infty$\end{document}.

2021, 41(1): i-ii doi: 10.3934/dcds.2020387 +[Abstract](1147) +[HTML](490) +[PDF](74.61KB)
Abstract:
2021, 41(3): 1415-1447 doi: 10.3934/dcds.2020323 +[Abstract](1236) +[HTML](437) +[PDF](509.61KB)
Abstract:

In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

2020, 40(12): i-ii doi: 10.3934/dcds.2020327 +[Abstract](1215) +[HTML](650) +[PDF](74.53KB)
Abstract:
2020, 40(8): 4907-4925 doi: 10.3934/dcds.2020205 +[Abstract](1370) +[HTML](486) +[PDF](815.51KB)
Abstract:

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [4] and [5], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

2020, 40(6): 3595-3627 doi: 10.3934/dcds.2020170 +[Abstract](1628) +[HTML](693) +[PDF](1079.38KB)
Abstract:

The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.

2020, 40(6): i-ii doi: 10.3934/dcds.2020171 +[Abstract](2284) +[HTML](576) +[PDF](78.36KB)
Abstract:
2020, 40(6): 3813-3836 doi: 10.3934/dcds.2020160 +[Abstract](1713) +[HTML](433) +[PDF](355.44KB)
Abstract:

Some reaction-diffusion models describing the cell polarity are proposed, where the system has two independent variables standing for the concentration of proteins in the membrane and the cytosol respectively. In this article we deal with such a polarity model consisting of one equation on a unit sphere and the other one in the ball inside the sphere. The two equations are coupled through a nonlinear boundary condition and the total mass is conserved. We investigate the linearized stability of a constant steady state and provide conditions under which a Turing type instability takes place, namely, the constant state is stable against spatially uniform perturbations on the sphere for all choices of diffusion rates, while unstable against nonuniform perturbations on the sphere as the diffusion coefficient of the equation on the sphere becomes small relative to the one in the ball.

2020, 40(6): 3837-3855 doi: 10.3934/dcds.2020169 +[Abstract](1429) +[HTML](370) +[PDF](374.36KB)
Abstract:

We study the following Neumann problem in one dimension,

\begin{document}$\left\{ {\begin{array}{*{20}{l}}\begin{array}{l}{u_t} = du'' + g(x){u^2}(1 - u)\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\0 \le u \le 1\quad {\rm{in}}\quad (0,1) \times (0,\infty ),\;\\u'(0,t) = u'(1,t) = 0\quad {\rm{in}}\quad (0,\infty ),\end{array}\end{array}} \right.$\end{document}

where \begin{document}$g$\end{document} changes sign in \begin{document}$(0, 1)$\end{document}. This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial stable steady state \begin{document}$U_d$\end{document} for \begin{document}$d$\end{document} sufficiently small. We show that \begin{document}$U_d$\end{document} is a unique nontrivial steady state under a condition \begin{document}$\int_{0}^1\, g(x)\, dx\geq 0$\end{document} and some other additional condition.

2020, 40(6): 3561-3570 doi: 10.3934/dcds.2020161 +[Abstract](1193) +[HTML](410) +[PDF](1908.79KB)
Abstract:

In 1979, Shigesada, Kawasaki and Teramoto [11] proposed a mathematical model with nonlinear diffusion, to study the segregation phenomenon in a two competing species community. In this paper, we discuss limiting systems of the model as the cross-diffusion rates included in the nonlinear diffusion tend to infinity. By formal calculation without rigorous proof, we obtain one limiting system which is a little different from that established in Lou and Ni [5].

2020, 40(6): 3857-3881 doi: 10.3934/dcds.2020128 +[Abstract](1596) +[HTML](398) +[PDF](504.48KB)
Abstract:

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as

\begin{document}$-F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle = \lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x),$\end{document}

for \begin{document}$i = 1, \cdots, n$\end{document}, in a bounded \begin{document}$C^{1, 1}$\end{document} domain \begin{document}$\Omega\subset \mathbb{R}^N$\end{document} with Dirichlet boundary conditions; here \begin{document}$n\geq 1$\end{document}, \begin{document}$\lambda \in \mathbb{R}$\end{document}, \begin{document}$c_{ij}, \, h_i \in L^\infty(\Omega)$\end{document}, \begin{document}$c_{ij}\geq 0$\end{document}, \begin{document}$M_i$\end{document} satisfies \begin{document}$0<\mu_1 I\leq M_i\leq \mu_2 I$\end{document}, and \begin{document}$F_i$\end{document} is an uniformly elliptic Isaacs operator.

We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

2020, 40(6): 3201-3214 doi: 10.3934/dcds.2020125 +[Abstract](1406) +[HTML](433) +[PDF](365.89KB)
Abstract:

The purpose of this paper is to study the solutions of

\begin{document}$\Delta u +K(x) e^{2u} = 0 \quad{\rm in}\;\; \mathbb{R}^2$\end{document}

with \begin{document}$K\le 0$\end{document}. We introduce the following quantities:

\begin{document}$\alpha_p(K) = \sup\left\{\alpha \in \mathbb{R}:\, \int_{ \mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$\end{document}

Under the assumption \begin{document}$({\mathbb H}_1)$\end{document}: \begin{document}$\alpha_p(K)> -\infty$\end{document} for some \begin{document}$p>1$\end{document} and \begin{document}$\alpha_1(K) > 0$\end{document}, we show that for any \begin{document}$0 < \alpha < \alpha_1(K)$\end{document}, there is a unique solution \begin{document}$u_\alpha$\end{document} with \begin{document}$u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big)$\end{document} at infinity and \begin{document}$\beta\in (0, \, \alpha_1(K)-\alpha)$\end{document}. Furthermore, we show an example \begin{document}$K_0 \leq 0$\end{document} such that \begin{document}$\alpha_p(K_0) = -\infty$\end{document} for any \begin{document}$p>1$\end{document} and \begin{document}$\alpha_1(K_0) > 0$\end{document}, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution \begin{document}$u_{\alpha_*}$\end{document} such that \begin{document}$u_{\alpha_*} -\alpha_*\ln|x| = O(1)$\end{document} at infinity for some \begin{document}$\alpha_* > 0$\end{document}, which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.

2020, 40(6): 3981-3995 doi: 10.3934/dcds.2020126 +[Abstract](1487) +[HTML](397) +[PDF](419.08KB)
Abstract:

For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [4]. This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for unbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation. Since pyramidal traveling fronts have been studied in many equations or systems, our method might be applicable to study axisymmetric traveling fronts in these equations or systems.

2020, 40(6): 3883-3907 doi: 10.3934/dcds.2020129 +[Abstract](1738) +[HTML](438) +[PDF](407.73KB)
Abstract:

We establish a general theory on the existence of fixed points and the convergence of orbits in order-preserving semi-dynamical systems having a certain mass conservation property (or, equivalently, a first integral). The base space is an ordered metric space and we do not assume differentiability of the system nor do we even require linear structure in the base space. Our first main result states that any orbit either converges to a fixed point or escapes to infinity (convergence theorem). This will be shown without assuming the existence of a fixed point. Our second main result states that the existence of one fixed point implies the existence of a continuum of fixed points that are totally ordered (structure theorem). This latter result, when applied to a linear problem for which \begin{document}$0$\end{document} is always a fixed point, automatically implies the existence of positive fixed points. Our result extends the earlier related works by Arino (1991), Mierczyński (1987) and Banaji-Angeli (2010) considerably with exceedingly simpler proofs. We apply our results to a number of problems including molecular motor models with time-periodic or autonomous coefficients, certain classes of reaction-diffusion systems and delay-differential equations.

2020, 40(6): 3291-3304 doi: 10.3934/dcds.2020127 +[Abstract](1491) +[HTML](437) +[PDF](382.66KB)
Abstract:

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [18]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

2020 Impact Factor: 1.392
5 Year Impact Factor: 1.610
2020 CiteScore: 2.2