# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

November 2002 , Volume 8 , Issue 4

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2002, 8(4): 815-833 doi: 10.3934/dcds.2002.8.815 +[Abstract](1170) +[PDF](239.0KB)
Abstract:
This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods of proof. The methods considered are fully numerical, in that they take into account the need to interpolate the approximations of the manifold between a set of discrete gridpoints. As all the discretisations are refined the approximations are shown to converge to the true manifold.
2002, 8(4): 835-849 doi: 10.3934/dcds.2002.8.835 +[Abstract](1229) +[PDF](202.9KB)
Abstract:
The spectrum of dimensions for Poincaré recurrences of Markov maps is obtained by constructing a sequence of approximating maps whose spectra are known to be solution of non-homogeneous Bowen equations. We prove that the spectrum of the Markov map also satisfies such an equation.
2002, 8(4): 851-864 doi: 10.3934/dcds.2002.8.851 +[Abstract](937) +[PDF](211.3KB)
Abstract:
We study the asymptotic behavior of solutions of the damped linear system $u_{t t}(t)+Au(t)+Bu_t(t)=0, t\geq 0$ in the context of Hilbert spaces. We present abstract theorems on the decay rate, moreover an adequate example is presented to illustrate these results.
2002, 8(4): 865-871 doi: 10.3934/dcds.2002.8.865 +[Abstract](981) +[PDF](159.6KB)
Abstract:
Using the technique of Adimurthy, F. Pacella and S.L. Yadava [1], we extend an uniqueness result for a class of non-autonomous semilinear equations in M.K. Kwong and Y. Li [8]. We also observe that combining the results of [1] with bifurcation theory, one can obtain a detailed picture of the global solution curve for a class of concave-convex nonlinearities.
2002, 8(4): 873-892 doi: 10.3934/dcds.2002.8.873 +[Abstract](1090) +[PDF](279.3KB)
Abstract:
We describe a global version of the KS regularization of the $n$-center problem on a closed 3-dimensional manifold. The regularized configuration manifold turns out to be 4 or 5 dimensional closed manifold depending on whether $n$ is even or odd. As an application, we show that the $n$ center problem in $S^3$ has positive topological entropy for $n\ge 5$ and energy greater than the maximum of the potential energy. The proof is based on the results of Gromov and Paternain on the topological entropy of geodesic flows. This paper is a continuation of [6], where global regularization of the $n$-center problem in $\mathbf R^3$ was studied.
2002, 8(4): 893-906 doi: 10.3934/dcds.2002.8.893 +[Abstract](1115) +[PDF](237.5KB)
Abstract:
In this paper we prove some regularity and uniqueness results for a class of nonlinear parabolic problems whose prototype is

$\partial_t u - \Delta_N u=\mu$ in $\mathcal D'(Q)$

$u=0$ on $]0,T[\times\partial \Omega$

$u(0)=u_0$ in $\Omega,$

where $Q$ is the cylinder $Q=(0,T)\times\Omega$, $T>0$, $\Omega\subset \mathbb R^n$, $N\ge 2$, is an open bounded set having $C^2$ boundary, $\mu\in L^1(0,T;M(\Omega))$ and $u_0$ belongs to $M(\Omega)$, the space of the Radon measures in $\Omega$, or to $L^1(\Omega)$. The results are obtained in the framework of the so-called grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

2002, 8(4): 907-930 doi: 10.3934/dcds.2002.8.907 +[Abstract](1221) +[PDF](273.8KB)
Abstract:
We study the existence of $2\pi$-periodic solutions for forced nonlinear oscillators at resonance, the nonlinearity being a bounded perturbation of a function deriving from an isochronous potential, i.e. a potential leading to free oscillations that all have the same period. The family of isochronous oscillators considered here includes oscillators with jumping nonlinearities, as well as oscillators with a repulsive singularity, to which a particular attention is paid. The existence results contain, as particular cases, conditions of Landesman-Lazer type. Even in the case of perturbed linear oscillators, they improve earlier results. Multiplicity and non-existence results are also given.
2002, 8(4): 931-937 doi: 10.3934/dcds.2002.8.931 +[Abstract](841) +[PDF](186.8KB)
Abstract:
We exhibit an open set of symplectic Anosov diffeomorphisms on which there are discrete "jumps" in the regularity of the unstable subbundle. It is either highly irregular almost everywhere ($C^\epsilon$ only on a negligible set) or better than $C^1$. In the latter case the Hölder exponent of the derivative is either about $\epsilon/2$ or almost 1.
2002, 8(4): 939-951 doi: 10.3934/dcds.2002.8.939 +[Abstract](1155) +[PDF](249.5KB)
Abstract:
We discuss estimates of the Hausdorff and fractal dimension of a global attractor for the semilinear wave equation

$u_{t t} +\delta u_t -\phi (x)\Delta u + \lambda f(u) = \eta (x), x \in \mathbb R^N, t \geq 0,$

with the initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x),$ where $N \geq 3$, $\delta >0$ and $(\phi (x))^{-1}:=g(x)$ lies in $L^{N/2}(\mathbb R^N)\cap L^\infty (\mathbb R^N)$. The energy space $\mathcal X_0=\mathcal D^{1,2}(\mathbb R^N) \times L_g^2(\mathbb R^N)$ is introduced, to overcome the difficulties related with the non-compactness of operators, which arise in unbounded domains. The estimates on the Hausdorff dimension are in terms of given parameters, due to an asymptotic estimate for the eigenvalues $\mu$ of the eigenvalue problem $-\phi(x)\Delta u=\mu u, x \in \mathbb R^N$.

2002, 8(4): 953-966 doi: 10.3934/dcds.2002.8.953 +[Abstract](1034) +[PDF](210.2KB)
Abstract:
We study Cauchy problems associated to partial differential equations with infinite delay where the history function is modified by an evolution family. Using sophisticated tools from semigroup theory such as evolution semigroups, extrapolation spaces, or the critical spectrum, we prove well-posedness and characterize the asymptotic behavior of the solution semigroup by an operator-valued characteristic equation.
2002, 8(4): 967-982 doi: 10.3934/dcds.2002.8.967 +[Abstract](881) +[PDF](216.9KB)
Abstract:
We study relaxation for optimal design problems in conductivity in the two-dimensional situation. To this end, we reformulate the optimal design problem in an equivalent way as a genuine vector variational problem, and then analyze relaxation of this new variational problem. Our main achievement is to explicitly compute the quasiconvexification of the involved density in this problem for some interesting cases. We think the method given here could be generalized to compute quasiconvex envelopes in other situations. We restrict attention to the two-dimensional case.
2002, 8(4): 983-994 doi: 10.3934/dcds.2002.8.983 +[Abstract](1055) +[PDF](191.1KB)
Abstract:
In this paper we construct a triangular map $F$ on $I^2$ which holds the following property. For each $[a,b]\subseteq I=[0,1]$, $a\leq b$, there exists $(p,q)\in I^2$ \ $I_0$ such that $\omega_F(p,q)=$ {0} $\times [a,b]\subset I_0$ where $I_0=${0}$\times I$. Moreover, for each $(p,q)\in I^{2}$, the set $\omega_F(p,q)$ is exactly {0} $\times J$ where $J\subset I$ is a compact interval degenerate or not. So, we describe completely the family $\mathcal W(F)=${$\omega_F(p,q):(p,q)\in I^2$} and establish $\mathcal W(F)$ as the set of all compact interval, degenerate or not, of $I_0$.
2002, 8(4): 995-1018 doi: 10.3934/dcds.2002.8.995 +[Abstract](870) +[PDF](319.2KB)
Abstract:
Consider the polynomial perturbations of Hamiltonian vector field

$X_\epsilon=(H_y+\epsilon f(x,y,\epsilon))\frac{\partial}{\partial x}+ (-H_x+\epsilon g(x,y,\epsilon))\frac{\partial}{\partial y},$

where the Hamiltonian $H(x,y)=\frac{1}{2}y^2+U(x)$ has one center and one cuspidal loop, $deg U(x)=4$. In present paper we find an upper bound for the number of zeros of the $k$th order Melnikov function $M_k(h)$ for arbitrary polynomials $f(x,y,\epsilon)$ and $g(x,y,\epsilon)$.

2002, 8(4): 1019-1024 doi: 10.3934/dcds.2002.8.1019 +[Abstract](995) +[PDF](164.7KB)
Abstract:
In the present paper, using the Leray-Schauder degree theory, we proved the existence of nontrivial solutions for p-Laplacian with a crossing nonlinearity.
2002, 8(4): 1025-1042 doi: 10.3934/dcds.2002.8.1025 +[Abstract](1040) +[PDF](227.8KB)
Abstract:
We establish the optimal rate of decay for the global solutions of some nonlinear partial differential equations with dissipation. We apply the well known Fourier splitting technique invented by Maria Schonbek in [1] -- [5] to achieve our goal.
2002, 8(4): 1043-1057 doi: 10.3934/dcds.2002.8.1043 +[Abstract](1089) +[PDF](215.2KB)
Abstract:
We consider the generalized Liénard system

$\frac{dx}{dt} = \frac{1}{a(x)}[h(y)-F(x)],$

$\frac{dy}{dt}= -a(x)g(x),\qquad\qquad\qquad\qquad\qquad$ (0.1)

where $a$ is a positive and continuous function on $R=(-\infty, \infty)$, and $F$, $g$ and $h$ are continuous functions on $R$. Under the assumption that the origin is a unique equilibrium, we obtain necessary and sufficient conditions for the origin of system (0.1) to be globally asymptotically stable by using a nonlinear integral inequality. Our results substantially extend and improve several known results in the literature.

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