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Discrete & Continuous Dynamical Systems - A

Open Access Articles

Editorial
Manuel del Pino, Shouchuan Hu and Juncheng Wei
2021, 41(1): i-ii doi: 10.3934/dcds.2020387 +[Abstract](343) +[HTML](139) +[PDF](74.61KB)
Abstract:
A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system
Masaru Hamano and Satoshi Masaki
2021, 41(3): 1415-1447 doi: 10.3934/dcds.2020323 +[Abstract](374) +[HTML](179) +[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.

Preface
Maciej Capiński, Núria Fagella, Michał Misiurewicz, Weixiao Shen, Benjamin Weiss and Krzysztof Ciepliński
2020, 40(12): ⅰ-ⅱ doi: 10.3934/dcds.2020327 +[Abstract](601) +[HTML](162) +[PDF](74.53KB)
Abstract:
Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation
Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa and Shoji Yotsutani
2020, 40(8): 4907-4925 doi: 10.3934/dcds.2020205 +[Abstract](585) +[HTML](225) +[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.

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems
Alexandra Köthe, Anna Marciniak-Czochra and Izumi Takagi
2020, 40(6): 3595-3627 doi: 10.3934/dcds.2020170 +[Abstract](749) +[HTML](240) +[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.

Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday
Chiun-Chuan Chen, Yuan Lou, Hirokazu Ninomiya, Peter Polacik and Xuefeng Wang
2020, 40(6): ⅰ-ⅱ doi: 10.3934/dcds.2020171 +[Abstract](1209) +[HTML](286) +[PDF](78.36KB)
Abstract:
Turing type instability in a diffusion model with mass transport on the boundary
Yoshihisa Morita and Kunimochi Sakamoto
2020, 40(6): 3813-3836 doi: 10.3934/dcds.2020160 +[Abstract](751) +[HTML](177) +[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.

Indefinite nonlinear diffusion problem in population genetics
Kimie Nakashima
2020, 40(6): 3837-3855 doi: 10.3934/dcds.2020169 +[Abstract](583) +[HTML](109) +[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.

On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates
Yukio Kan-On
2020, 40(6): 3561-3570 doi: 10.3934/dcds.2020161 +[Abstract](573) +[HTML](157) +[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].

A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth
Gabrielle Nornberg, Delia Schiera and Boyan Sirakov
2020, 40(6): 3857-3881 doi: 10.3934/dcds.2020128 +[Abstract](764) +[HTML](146) +[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.

On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature
Huyuan Chen, Dong Ye and Feng Zhou
2020, 40(6): 3201-3214 doi: 10.3934/dcds.2020125 +[Abstract](643) +[HTML](161) +[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.

Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations
Masaharu Taniguchi
2020, 40(6): 3981-3995 doi: 10.3934/dcds.2020126 +[Abstract](633) +[HTML](156) +[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.

Convergence and structure theorems for order-preserving dynamical systems with mass conservation
Toshiko Ogiwara, Danielle Hilhorst and Hiroshi Matano
2020, 40(6): 3883-3907 doi: 10.3934/dcds.2020129 +[Abstract](767) +[HTML](177) +[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.

On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen and Chih-Her Chen
2020, 40(6): 3291-3304 doi: 10.3934/dcds.2020127 +[Abstract](608) +[HTML](170) +[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.

Boundary spike of the singular limit of an energy minimizing problem
Xinfu Chen, Huiqiang Jiang and Guoqing Liu
2020, 40(6): 3253-3290 doi: 10.3934/dcds.2020124 +[Abstract](654) +[HTML](172) +[PDF](461.47KB)
Abstract:

In this paper, we consider the singular limit of an energy minimizing problem which is a semi-limit of a singular elliptic equation modeling steady states of thin film equation with both Van der Waals force and Born repulsion force. We show that the singular limit of energy minimizers is a Dirac mass located on the boundary point with the maximum curvature.

On space-time periodic solutions of the one-dimensional heat equation
Dong-Ho Tsai and Chia-Hsing Nien
2020, 40(6): 3997-4017 doi: 10.3934/dcds.2020037 +[Abstract](892) +[HTML](234) +[PDF](444.59KB)
Abstract:

We look for solutions \begin{document}$ u\left( x,t\right) $\end{document} of the one-dimensional heat equation \begin{document}$ u_{t} = u_{xx} $\end{document} which are space-time periodic, i.e. they satisfy the property

\begin{document}$ u\left( x+a,t+b\right) = u\left( x,t\right) $\end{document}

for all \begin{document}$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $\end{document} and derive their Fourier series expansions. Here \begin{document}$ a\geq0,\ b\geq 0 $\end{document} are two constants with \begin{document}$ a^{2}+b^{2}>0. $\end{document} For general equation of the form \begin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $\end{document} where \begin{document}$ A,\ B $\end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \begin{document}$ B>0 $\end{document} and is given by a linear combination of \begin{document}$ \cos\left( \sqrt{B}\left( x+At\right) \right) $\end{document} and \begin{document}$ \sin\left( \sqrt{B}\left( x+At\right) \right). $\end{document}

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small
Xueli Bai and Fang Li
2020, 40(6): 3075-3092 doi: 10.3934/dcds.2020035 +[Abstract](995) +[HTML](289) +[PDF](384.66KB)
Abstract:

In this paper, we study the global dynamics of a general \begin{document}$ 2\times 2 $\end{document} competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper extends the work in [3], where Lotka-Volterra competition models with symmetric nonlocal operators are considered, to more general competition models with nonsymmetric operators.

Variational and operator methods for Maxwell-Stokes system
Xing-Bin Pan
2020, 40(6): 3909-3955 doi: 10.3934/dcds.2020036 +[Abstract](1029) +[HTML](336) +[PDF](545.81KB)
Abstract:

In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.

Asymptotic population abundance of a two-patch system with asymmetric diffusion
Mengting Fang, Yuanshi Wang, Mingshu Chen and Donald L. DeAngelis
2020, 40(6): 3411-3425 doi: 10.3934/dcds.2020031 +[Abstract](812) +[HTML](279) +[PDF](622.93KB)
Abstract:

This paper considers a two-patch system with asymmetric diffusion rates, in which exploitable resources are included. By using dynamical system theory, we exclude periodic solution in the one-patch subsystem and demonstrate its global dynamics. Then we exhibit uniform persistence of the two-patch system and demonstrate uniqueness of the positive equilibrium, which is shown to be asymptotically stable when the diffusion rates are sufficiently large. By a thorough analysis on the asymptotic population abundance, we demonstrate necessary and sufficient conditions under which the asymmetric diffusion rates can lead to the result that total equilibrium population abundance in heterogeneous environments is larger than that in heterogeneous/homogeneous environments with no diffusion, which is not intuitive. Our result extends previous work to the situation of asymmetric diffusion and provides new insights. Numerical simulations confirm and extend our results.

Refined regularity and stabilization properties in a degenerate haptotaxis system
Michael Winkler and Christian Stinner
2020, 40(6): 4039-4058 doi: 10.3934/dcds.2020030 +[Abstract](709) +[HTML](218) +[PDF](401.32KB)
Abstract:

We consider the degenerate haptotaxis system

\begin{document}$ \left\{ \begin{array}{l} u_t = (d(x)u)_{xx} - (d(x)uw_x )_x, \\ w_t = -ug(w), \end{array} \right. $\end{document}

endowed with no-flux boundary conditions in a bounded open interval \begin{document}$ \Omega \subset \mathbb{R} $\end{document}. It was proposed as a basic model for haptotactic migration in heterogeneous environments. If the diffusion is degenerate in the sense that \begin{document}$ d $\end{document} is non-negative, has a non-empty zero set and satisfies \begin{document}$ \int_\Omega \frac{1}{d} <\infty $\end{document}, then it has been shown in [12] under appropriate assumptions on the initial data that the system has a global generalized solution satisfying in particular \begin{document}$ u(\cdot,t) \rightharpoonup \frac{\mu_\infty}{d} $\end{document} weakly in \begin{document}$ L^1 (\Omega) $\end{document} as \begin{document}$ t \to \infty $\end{document} for some positive constant \begin{document}$ \mu_\infty $\end{document}.

We now prove that under the additional restriction \begin{document}$ \int_\Omega \frac{1}{d^2} <\infty $\end{document} we have the strong convergence \begin{document}$ u(\cdot,t)\to \frac{\mu_\infty}{d} $\end{document} in \begin{document}$ L^p (\Omega) $\end{document} as \begin{document}$ t \to \infty $\end{document} for any \begin{document}$ p \in (1,2) $\end{document}. In addition, with the same restriction on \begin{document}$ d $\end{document} we obtain improved regularity properties of \begin{document}$ u $\end{document}, for instance \begin{document}$ du \in L^\infty ((0,\infty); L^p(\Omega)) $\end{document} for any \begin{document}$ p \in (1,\infty) $\end{document}.

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