# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

July 1999 , Volume 5 , Issue 3

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1999, 5(3): 457-472 doi: 10.3934/dcds.1999.5.457 +[Abstract](2018) +[PDF](251.8KB)
Abstract:
Consider a propagator defined on a Banach space whose norm satisfies an appropriate exponential bound. To this operator is added a bounded operator which is relatively smoothing in the sense of Vidav. The location of the essential spectrum of the perturbed propagator is then estimated. An application to kinetic theory is given for a system of particles that interact both through collisions and through their charges.
1999, 5(3): 473-488 doi: 10.3934/dcds.1999.5.473 +[Abstract](1832) +[PDF](219.2KB)
Abstract:
In this paper we get a lower bound independent of $\delta$ on the life-span of classical solutions to the following Cauchy problem by using the global iteration method

$\delta u_{t t}-\Delta u +u_t = F(u, \nabla u),$

$t = 0 : u = \epsilon u_0(x), u_t = \epsilon u_1(x),$

where $\delta$ and $\epsilon$ are small positive parameters. Moreover, we consider the related singular perturbated problem as $\delta\to 0$ and show that the perturbated term $\delta u_{t t}$ has an appreciable effect only for a short times.

1999, 5(3): 489-493 doi: 10.3934/dcds.1999.5.489 +[Abstract](1807) +[PDF](138.9KB)
Abstract:
In this paper, we use Lyapunov-Schmidt method and Morse theory to study semilinear elliptic boundary value problems with resonance at infinity, and get new multiple solutions theorems.
1999, 5(3): 495-514 doi: 10.3934/dcds.1999.5.495 +[Abstract](1636) +[PDF](275.8KB)
Abstract:
In this paper we develop a theory of k-adic expansion of an integral aritmethic function. Applying this formal language to Lefschetz numbers, or fixed point indices, of iterations of a given map we reformulate or reprove earlier results of Babienko-Bogatyj, Bowszyc, Chow-Mallet-Paret and Franks. Also we give a new characterization of a sequence of Lefschetz numbers of iterations of a map $f$: For a smooth transversal map we get more refined version of Matsuoka theorem on parity of number of orbits of a transversal map. Finally, for any $C^1$-map we show the existence of infinitely many prime periods provided the sequence of Lefschetz numbers of iterations is unbounded.
1999, 5(3): 515-528 doi: 10.3934/dcds.1999.5.515 +[Abstract](1475) +[PDF](234.0KB)
Abstract:
In this paper we improve a general theorem of O.A. Ladyzhenskaya on the dimension of compact invariant sets in Hilbert spaces. Then we use this result to prove that the Hausdorff and fractal dimensions of global compact attractors of differential inclusions and reaction-diffusion equations are finite.
1999, 5(3): 529-534 doi: 10.3934/dcds.1999.5.529 +[Abstract](1371) +[PDF](171.1KB)
Abstract:
none
1999, 5(3): 535-552 doi: 10.3934/dcds.1999.5.535 +[Abstract](1941) +[PDF](258.6KB)
Abstract:
We study the scaling function of a $C^{1+h}$ expanding circle endomorphism. We find necessary and sufficient conditions for a Hölder continuous function on the dual symbolic space to be realized as the scaling function of a $C^{1+h}$ expanding circle endomorphism. We further represent the Teichmüller space of $C^{1+h}$ expanding circle endomorphisms by the space of Hölder continuous functions on the dual symbolic space satisfying our necessary and sufficient conditions and study the completion of this Teichmüller space in the universal Teichmüller space.
1999, 5(3): 553-568 doi: 10.3934/dcds.1999.5.553 +[Abstract](1710) +[PDF](270.2KB)
Abstract:
In this paper we prove the existence of a solution for a class of noncoercive Cauchy problems whose prototype is the boundary value problem

$\frac{\partialu}{\partial t}-$ div$(|Du|^{p-2}Du) + B(x, t)\cdot |Du|^{\gamma-1}Du = f$ in $\Omega_T$,

$u(x, t)=0$ on ­$\Omega\times (0, T),$

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

under suitable hypotheses on the data.

1999, 5(3): 569-584 doi: 10.3934/dcds.1999.5.569 +[Abstract](1668) +[PDF](272.8KB)
Abstract:
We regard second order systems of the form $\ddot q = \nabla_qW(q, t), t\in \mathbb R, q \in \mathbb R^N,$ where $W(q, t)$ is $\mathbb Z^N$ periodic in $q$ and almost periodic in $t$. Variational arguments are used to prove the existence of heteroclinic solutions joining almost periodic solutions to the system.
1999, 5(3): 585-598 doi: 10.3934/dcds.1999.5.585 +[Abstract](1343) +[PDF](223.3KB)
Abstract:
We consider a thermo-elastic plate equation with rotational forces [Lagnese.1] and with coupled hinged mechanical/Neumann thermal boundary conditions (B.C.). We give a sharp result on the Neumann trace of the mechanical velocity, which is "$\frac{1}{2}$" sharper in the space variable than the result than one would obtain by a formal application of trace theory on the optimal interior regularity. Two proofs by energy methods are given: one which reduces the analysis to sharp wave equation's regularity theory; and one which analyzes directly the corresponding Kirchoff elastic equation. Important implications of this result are noted.
1999, 5(3): 599-616 doi: 10.3934/dcds.1999.5.599 +[Abstract](1805) +[PDF](243.8KB)
Abstract:
Recent work has shown that in the setting of continuous maps on a locally compact metric space the spectrum of the Conley index can be used to conclude that the dynamics of an invariant set is at least as complicated as that of full shift dynamics on two symbols, that is, a horseshoe. In this paper, one considers which spectra are possible and then produce examples which clearly delineate which spectral conditions do or do not allow one to conclude the existence of a horseshoe.
1999, 5(3): 617-630 doi: 10.3934/dcds.1999.5.617 +[Abstract](1579) +[PDF](1985.7KB)
Abstract:
An orbital Conley index for non-invariant compact sets of discrete-time dynamical systems is introduced. The construction of this new index uses an algebraic reduction process inspired from Leray. Applications to detection of periodic orbits and chaos are presented.
1999, 5(3): 631-638 doi: 10.3934/dcds.1999.5.631 +[Abstract](1551) +[PDF](172.9KB)
Abstract:
In this paper, we prove the zero diffusion limit of 2-D incompressible Navier- Stokes equations with $L^1(\mathcal R^2)$ initial vorticity is still a weak solution of the corresponding Euler equations.
1999, 5(3): 639-650 doi: 10.3934/dcds.1999.5.639 +[Abstract](1262) +[PDF](225.8KB)
Abstract:
Let $M$ be a closed connected $C^\infty$ Riemannian manifold whose geodesic flow $\phi$ is Anosov. Let $\theta$ be a smooth 1-form on $M$. Given $\lambda\in \mathbb R$ small, let $h_{E L}(\lambda)$ be the topological entropy of the Euler-Lagrange flow of the Lagrangian

$L_\lambda (x, v) =\frac{1}{2}|v|^2_x-\lambda\theta_x(v),$

and let $h_F(\lambda)$ be the topological entropy of the geodesic flow of the Finsler metric,

$F_\lambda(x, v) = |v|_x-\lambda\theta_x(v),$

We show that $h_{E L}''(0) + h''_F(0) = h^2$Var$(\theta)$, where Var$(\theta)$ is the variance of $\theta$ with respect to the measure of maximal entropy of $\phi$ and $h$ is the topological entropy of $\phi$. We derive various consequences from this formula.

1999, 5(3): 651-662 doi: 10.3934/dcds.1999.5.651 +[Abstract](1282) +[PDF](221.1KB)
Abstract:
Of concern is the differentiability of the propagators of higher order Cauchy problem

$u^{(n)}(t) +\sum_{k=0}^{n-1}A_ku^{(k)}(t)=0, t\geq 0,$

$u^{(k)}(0) = u_k, 0\leq k\leq n-1, where$A_0, A_1,\ldots, A_{n-1}$are densely defined closed linear operators on a Banach space. A Pazy-type characterization of the infinitely differentiable propagators of the Cauchy problem is obtained. Moreover, two related sufficient conditions are given. 1999, 5(3): 663-683 doi: 10.3934/dcds.1999.5.663 +[Abstract](1831) +[PDF](3859.2KB) Abstract: We prove existence and uniquences of positive solutions of an age-structured population equation of McKendrick type with spatial diffusion in$L^1$. The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [19], [24], [30], and [31] to the non-autonomous situation. We use the theory of evolution semigroups and extrapolation spaces. 1999, 5(3): 685-695 doi: 10.3934/dcds.1999.5.685 +[Abstract](2473) +[PDF](220.3KB) Abstract: In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type:$iu_t + u_{x x} = \mathcal N(u, \bar u, u_x, \bar u_x), \quad t \in \mathbf R,\ x\in \mathbf R;\quad u(0, x) = u_0(x),\ x\in \mathbf R,\qquad$(A) where$\mathcal N(u, \bar u, u_x, \bar u_x) = K_1|u|^2u+K_2|u|^2u_x +K_3u^2\bar u_x +K_4|u_x|^2u+K_5\bar uu_x^2 +K_6|u_x|^2u_x$, the functions$K_j = K_j (|u|^2)$,$K_j(z)\in C^\infty ([0, \infty))$. If the nonlinear terms$\mathcal N =\frac{\bar{u} u_x^2}{1+|u|^2}$, then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity$\mathcal N$depends both on$u_x$and$\bar u_x$. We prove that if the initial data$u_0\in H^{3, \infty}$and the norms$||u_0||_{3, l}$are sufficiently small for any$l\in N$, (when$\mathcal N$depends on$\bar u_x$), then for some time$T > 0$there exists a unique solution$u\in C^\infty ([-T, T]$\$\{0\};\ C^\infty(\mathbb R))$of the Cauchy problem (A). Here$H^{m, s} = \{\varphi \in \mathbf L^2;\ ||\varphi||_{m, s}<\infty \}$,$||\varphi||_{m, s}=||(1+x^2)^{s/2}(1-\partial_x^2)^{m/2}\varphi||_{\mathbf L^2}, \mathbf H^{m, \infty}=\cap_{s\geq 1} H^{m, s}.\$

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