# American Institue of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2005 , Volume 12 , Issue 5

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Guillaume Bal and  Lenya Ryzhik
2005, 12(5): 793-815 doi: 10.3934/dcds.2005.12.793 +[Abstract](24) +[PDF](325.4KB)
Abstract:
We analyze the refocusing properties of time reversed waves that propagate in two different media during the forward and backward stages of a time-reversal experiment. We consider two regimes of wave propagation modeled by the paraxial wave equation with a smooth random refraction coefficient and the Itô-Schrödinger equation, respectively. In both regimes, we rigorously characterize the refocused signal in the high frequency limit and show that it is statistically stable, that is, independent of the realizations of the two media. The analysis is based on a characterization of the high frequency limit of the Wigner transform of two fields propagating in different media.
The refocusing quality of the backpropagated signal is determined by the cross correlation of the two media. When the two media decorrelate, two distinct de-focusing effects are observed. The first one is a purely absorbing effect due to the loss of coherence at a fixed frequency. The second one is a phase modulation effect of the refocused signal at each frequency. This causes de-focusing of the backpropagated signal in the time domain.
2005, 12(5): 817-826 doi: 10.3934/dcds.2005.12.817 +[Abstract](30) +[PDF](188.8KB)
Abstract:
We discuss bifurcation problems for weakly nonlinear elliptic elliptic equations involving a pth power term where p is larger than 1 but close to 1.We show that in a number of cases one can obtain rather complete information about the structure of the set of positive solutions.These results apply to any bounded domain.
Klaudiusz Wójcik and  Piotr Zgliczyński
2005, 12(5): 827-852 doi: 10.3934/dcds.2005.12.827 +[Abstract](31) +[PDF](307.4KB)
Abstract:
We show that if an ordinary differential equation $x'=f(x)$, where $x\in \mathbb R^n$ and $f \in \mathcal C^1$, has a topological horseshoe, then the corresponding delay equation $x'(t)=f(x(t-h))$ for small $h >0$ also has a topological horseshoe, i.e. symbolic dynamics and an infinite number of periodic orbits. A method of computation of $h$ is given in terms of topological properties of solutions of differential inclusion $x'(t) \in f(x(t)) + \bar B(0,\delta)$.
2005, 12(5): 853-880 doi: 10.3934/dcds.2005.12.853 +[Abstract](26) +[PDF](398.6KB)
Abstract:
We use techniques of tube-log Riemann surfaces due to R.Pérez-Marco to construct a hedgehog containing smooth $C^\infty$ combs. The hedgehog is a common hedgehog for a family of commuting non-linearisable holomorphic maps with a common indifferent fixed point. The comb is made up of smooth curves, and is transversally bi-Hölder regular.
2005, 12(5): 881-886 doi: 10.3934/dcds.2005.12.881 +[Abstract](77) +[PDF](176.3KB)
Abstract:
In this paper we consider the regularity criteria for the solution to the 3D MHD equations. It is proved that if the gradient of the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 2$ or the velocity field belongs to $L^{\alpha,\gamma}$ with $2/\alpha+3/\gamma \leq 1$ on $[0,T]$, then the solution remains smooth on $[0,T]$. The significance is that there are no restriction on the magnetic field. Moreover, the norms $||\nabla u||_{L^{\alpha,\gamma}}$ and $\|\|u\|\|_{L^{\alpha,\gamma}}$ are scaling dimension zero for $2/\alpha+3/\gamma=2$ and $2/\alpha+3/\gamma=1$ respectively.
Víctor Guíñez and  Eduardo Sáez
2005, 12(5): 887-904 doi: 10.3934/dcds.2005.12.887 +[Abstract](22) +[PDF](418.6KB)
Abstract:
We complete the local study of rank--2 singular points of positive quadratic differential forms on oriented two--dimensional manifolds. We associate to each positive quadratic differential form $\omega$ defined on an oriented two--dimensional manifold $M$ two transversal one--dimensional foliations $f_1(\omega)$ and $f_2(\omega)$ with common set of singular points. This study was begun in [Gut-Gui] for a generic class of singularities called simple, and continued in [Gui-Sa] for those non--simple rank--2 singular points called of type C. Taking into account the classification of [Gui3], the only rank--2 singular points which remain to be studied are those of type E($\lambda$), for $\lambda\geq 1$. We undertake the local study of the remaining case under a non--flatness condition on the positive quadratic differential form at the singular point.
Roberto Alicandro , Andrea Braides and  Marco Cicalese
2005, 12(5): 905-928 doi: 10.3934/dcds.2005.12.905 +[Abstract](20) +[PDF](334.9KB)
Abstract:
We study necessary and sufficient conditions for the lower-semicontinuity of one-dimensional energies defined on ($BV$ and) $SBV$ of the model form $F(u)=$sup$f(u')\vee$sup$g([u])$, and prove a relaxation theorem. We apply these results to the study of problems with Dirichlet boundary conditions, highlighting a complex behaviour of solutions. We draw a comparison with the parallel theory for integral energies on $SBV$.
Bernard Brighi and  Tewfik Sari
2005, 12(5): 929-948 doi: 10.3934/dcds.2005.12.929 +[Abstract](46) +[PDF](1833.9KB)
Abstract:
We introduce blowing-up coordinates to study the autonomous third order nonlinear differential equation : $f'''+\frac{m+1}{2}ff''-m f'^2=0$ on $(0,\infty)$, subject to the boundary conditions $f(0)=a\in\mathbb R$, $f'(0)=1$ and $f'(t)\to 0$ as $t\to\infty$. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as free convection in porous medium or flow adjacent to a stretching wall. We study the corresponding plane dynamical systems and apply the results obtained to the original boundary value problem, in order to solve questions for which direct approach fails.
Fuqin Sun and  Mingxin Wang
2005, 12(5): 949-958 doi: 10.3934/dcds.2005.12.949 +[Abstract](47) +[PDF](208.7KB)
Abstract:
In this paper, we focus on the Cauchy problems of nonlinear strongly damped hyperbolic equations and systems. We give some conditions on the non-existence of global solutions.
Jacek Banasiak and  Marcin Moszyński
2005, 12(5): 959-972 doi: 10.3934/dcds.2005.12.959 +[Abstract](40) +[PDF](283.9KB)
Abstract:
In [8] the authors proved that a linear dynamical system $\mathcal T$ on a Banach space $X$ is topologically chaotic if there exists a selection of eigenvectors of the generator of $\mathcal T$, that is analytic in some open set of a complex plane that meets the imaginary axis, and such that a non-degeneracy condition holds. In this paper we show that if we drop the last assumption, then $\mathcal T$ is still chaotic albeit in a possibly smaller, but still infinite-dimensional, $\mathcal T$-invariant subspace of $X$. Such kind of chaotic behaviour we shall call subspace chaos. We also present criteria that allow to rule out subspace chaos in certain dynamical systems and discuss simple but instructive examples where these criteria are applied to the birth, as well as the death, type systems of population dynamics.
2005, 12(5): 973-982 doi: 10.3934/dcds.2005.12.973 +[Abstract](22) +[PDF](223.4KB)
Abstract:
We prove $C^\infty$ smoothness and uniform exponential decay for $H^1$-localized solutions moving to the right of the generalized BBM equation. For that purpose we use a monotonicity property for solution which are not necessarily close to solitary waves.
2005, 12(5): 983-996 doi: 10.3934/dcds.2005.12.983 +[Abstract](41) +[PDF](235.1KB)
Abstract:
In this paper, we are concerned with the following boundary value problem for Hamiltonian systems

$J\dot u(t)+\nabla H(t,u(t))=0$ a.e. on $[0,T]$

$u(0)=u(T)$

where the function $H:[0,T]\times \mathbb R^{2N}\rightarrow \mathbb R$ is called Hamiltonian. Our attention will be focused upon the case in which the Hamiltonian H, besides being measurable on $t\in[0,T]$, is convex and continuously differentiable with respect to $u\in \mathbb R^{2N}$. Our basic assumption is that the Hamiltonian $H$ satisfies the following growth condition:
Let $1 < p < 2$ and $q=\frac{p}{p-1}$. There exist positive constants $\alpha,\delta$ and functions $\beta,\gamma \in L^q(0,T;\mathbb R^+)$ such that

$\delta|u|-\beta(t)\leq H(t,u)\leq\frac{\alpha}{q}|u|^q+\gamma(t),$

for all $u\in \mathbb R^{2N}$ and a.e. $t\in[0,T]$. Our main result assures that under suitable bounds on $\alpha,\delta$ and the functions $\beta,\gamma$, the problem above has at least a solution that belongs to $W_T^{1,p}$. Such a solution corresponds, in the duality, to a function that minimizes the dual action restricted to a subset of $\tilde{W}_T^{1,p}=${$v\in W_T^{1,p}: \int_0^{ T} v(t) dt=0$}.

2005, 12(5): 997-1018 doi: 10.3934/dcds.2005.12.997 +[Abstract](107) +[PDF](241.3KB)
Abstract:
In this paper we discuss the existence and the exponential behaviour of the solutions to a 2D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^k+b_f,$ $a>0,b_f\geq 0,k\geq 2.$ $\tau (t)$ is a differentiable function with $0\leq \tau (t)\leq r, r>0,\frac{d}{dt}\tau (t)\leq M<1,$ $M$ a constant. We show the relations between the kinematic viscosity $\nu ,$ time delay $r>0$ and $\lambda_1, a, b_{f}, k, M$ play an important role. Furthermore, we consider the exponential behaviour of the strong solutions to a 3D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^2+b_f,$ $a>0,b_f\geq 0.$ We extend Corollary 64.5[11]. Furthermore we discuss the existence of a periodic solution.
S. Gatti , M. Grasselli , V. Pata and  M. Squassina
2005, 12(5): 1019-1029 doi: 10.3934/dcds.2005.12.1019 +[Abstract](25) +[PDF](235.1KB)
Abstract:
We consider a family of phase-field systems with memory effects in the temperature $\vartheta$, depending on a parameter $\omega\geq 0$. Setting the problems in a suitable phase-space accounting for the past history of $\vartheta$, we prove the existence of a family of exponential attractors $\mathcal E_\omega$ which is robust as $\omega\to 0$.

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