# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2005 , Volume 12 , Issue 2

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2005, 12(2): 185-192 doi: 10.3934/dcds.2005.12.185 +[Abstract](2222) +[PDF](190.8KB)
Abstract:
We consider a strongly coupled nonlinear parabolic system which arises from population dynamics in $N$-dimensional $(N\geq 1)$ domains. We establish global existence of classical solutions under certain restrictions on diffusion coefficients, self-diffusion coefficients and cross-diffusion coefficients for both species.
2005, 12(2): 193-212 doi: 10.3934/dcds.2005.12.193 +[Abstract](3242) +[PDF](259.7KB)
Abstract:
We study entire solutions of a scalar reaction-diffusion equation of 1-space dimension. Here the entire solutions are meant by solutions defined for all $(x,t)\in\mathbb R^2$. Assuming that the equation has traveling front solutions and using the comparison argument, we prove the existence of entire solutions which behave as two fronts coming from the both sides of $x$-axis. A key idea for the proof of the main results is to characterize the asymptotic behavior of the solutions as $t\to-\infty$ in terms of appropriate subsolutions and supersolutions. This argument can apply not only to a general bistable reaction-diffusion equation but also to the Fisher-KPP equation. We also extend our argument to the Fisher-KPP equation with discrete diffusion.
2005, 12(2): 213-232 doi: 10.3934/dcds.2005.12.213 +[Abstract](1919) +[PDF](236.3KB)
Abstract:
The uniform forward global attractors of periodic systems and the comparison of the dynamics between asymptotically periodic non-autonomous dynamical systems and their corresponding limiting periodic processes are investigated. The existence and Lyapunov stability of such attractors are established and a Lyapunov functional charecterization of uniform forward global attractors of non-autonomous periodic processes is presented. The results are illustrated with examples of ordinary and delay differential equations.
2005, 12(2): 233-242 doi: 10.3934/dcds.2005.12.233 +[Abstract](2033) +[PDF](188.5KB)
Abstract:
An interaction equations of the complex scalar nucleon field and real scalar meson field is considered. we show that the Cauchy problem of the Klein-Gordon-Schrödinger system

$iu_{t}+u_{x x}=-uv,$

$v_{t t}-v_{x x}+v=|u|^2,$

$u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$

is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.

2005, 12(2): 243-250 doi: 10.3934/dcds.2005.12.243 +[Abstract](1821) +[PDF](185.1KB)
Abstract:
Linear systems with partially uncertain time-dependent coefficients naturally arise in various areas of applied sciences. The Lyapunov function method became the prevailing approach to stability analysis of these systems, where it implies sufficient conditions for asymptotic stability. However, this method is not applicable to Hamiltonian systems, because they are at the utmost neutrally (but not asymptotically) stable. This paper describes stability analysis of linear Hamiltonian systems with partially uncertain periodic coefficients resulting in a generalization of a famous Yakubovich's stability theorem which significantly reduces its computational burden. We assume that the Hamiltonian is described as a sum of known and uncertain matrices, and for the latter only bilateral bounds are known. We propose a definition of such a system's stability and derive a necessary and sufficient stability criterion determined by positions of the multipliers of the corresponding marginal systems. Systems with a more general definition of uncertain matrices are also considered.
2005, 12(2): 251-282 doi: 10.3934/dcds.2005.12.251 +[Abstract](1855) +[PDF](291.3KB)
Abstract:
This paper concerns the time asymptotic behavior toward large rarefaction waves of the solution to general systems of $2\times 2$ hyperbolic conservation laws with positive viscosity coefficient $B(u)$

$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad$ ($*$)

$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow \pm\infty.$

Assume that the corresponding Riemann problem

$u_t+F(u)_x=0,$

$u(0,x)=u^r_0(x)=u_-,\quad x<0, and u_+,\quad x>0$

can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is a small perturbation of an approximate rarefaction wave constructed in Section 2, then we show that the Cauchy problem ($*$) admits a unique global smooth solution $u(t,x)$ which tends to $u^r(t,x)$ as the $t$ tends to infinity. Here, we do not require $|u_+ - u_-|$ to be small and thus show the convergence of the corresponding global smooth solutions to strong rarefaction waves for $2\times 2$ viscous conservation laws.

2005, 12(2): 283-302 doi: 10.3934/dcds.2005.12.283 +[Abstract](2058) +[PDF](301.5KB)
Abstract:
We continue our investigation of versality for parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds. In this paper, we consider RFDEs equivariant with respect to the action of a compact Lie group. In a previous paper (Buono and LeBlanc, J. Diff. Eqs., 193 , 307-342 (2003)), we have studied this question in the general case (i.e. no a priori restrictions on the RFDE). When studying the question of versality in the equivariant context, it is natural to want to restrict the range of possible unfoldings to include only those which share the same symmetries as the original RFDE, and so our previous results do not immediately apply. In this paper, we show that with appropriate projections, our previous results on versal unfoldings of linear RFDEs can be adapted to the case of linear equivariant RFDEs. We illustrate our theory by studying the linear equivariant unfoldings at double Hopf bifurcation points in a $\mathbb D_3$-equivariant network of coupled identical neurons modeled by delay-differential equations due to delays in the internal dynamics and coupling.
2005, 12(2): 303-313 doi: 10.3934/dcds.2005.12.303 +[Abstract](1867) +[PDF](168.9KB)
Abstract:
We establish the homogenization of the boundary value problem of a second order differential equation. It generates nonlocal effect. The eigenfunction expansion and Fredholm integral equation are exploited to obtain a characterization of the kernel while in the space independent case the Young measure is applied to obtain the explicit formula of the kernel.
2005, 12(2): 315-322 doi: 10.3934/dcds.2005.12.315 +[Abstract](2292) +[PDF](194.6KB)
Abstract:
The strong instability of ground state standing wave solutions $e^{i\omega t}\phi_{\omega}(x)$ for nonlinear Klein-Gordon equations has been known only for the case $\omega=0$. In this paper we prove the strong instability for small frequency $\omega$.
2005, 12(2): 323-346 doi: 10.3934/dcds.2005.12.323 +[Abstract](2317) +[PDF](276.7KB)
Abstract:
This is the second of two papers on boundary optimal control problems with linear state equation and convex cost arising from boundary control of PDEs and the the associated Hamilton--Jacobi--Bellman equation. In the first paper we studied necessary and sufficient conditions of optimality (Pontryagin Maximum Principle). In this second paper we will apply Dynamic Programming to show that the value function of the problem is a solution of an integral version of the HJB equation, and moreover that it is the pointwise limit of classical solutions of approximating equations.
2005, 12(2): 347-354 doi: 10.3934/dcds.2005.12.347 +[Abstract](2582) +[PDF](197.6KB)
Abstract:
Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we study more general integral equation

$u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy. We establish regularity, radial symmetry, and monotonicity of the solutions. We also consider subcritical cases, super critical cases, and singular solutions in all cases; and obtain qualitative properties for these solutions. 2005, 12(2): 355-362 doi: 10.3934/dcds.2005.12.355 +[Abstract](2951) +[PDF](188.4KB) Abstract: In this paper we consider the shadowing property for$C^1\$ random dynamical systems. We first define a type of hyperbolicity on the full measure invariant set which is given by Oseledec's multiplicative ergodic theorem, and then prove that the system has the "Lipschitz" shadowing property on it.
2005, 12(2): 363-375 doi: 10.3934/dcds.2005.12.363 +[Abstract](2072) +[PDF](238.2KB)
Abstract:
The goal of this paper is to analyze the dynamics and the Devaney-chaotic behavior of some classes of real rational functions. A key element is the description of the pull-backs of the set of points where the denominator has a zero. The kneading theory developed by Milnor and Thurston is applied to this set in order to establish topological conjugacy between some of these classes.

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