# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems

July 1997 , Volume 3 , Issue 3

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1997, 3(3): 305-316 doi: 10.3934/dcds.1997.3.305 +[Abstract](2515) +[PDF](223.4KB)
Abstract:
We study the local Cauchy problem for the semilinear parabolic equations

$\partial _t U-\Delta U=P(D)F(U), \quad (t,x) \in [0,T[ \times \mathbb{R}^n$

with initial data in Sobolev spaces of fractional order $H^s_p(\mathbb{R}^n)$. The techniques that we use allow us to consider measures but also distributions as initial data ($s<0$). We also prove some smoothing effects and $L^q([0,T[,L^p)$ estimates for the solutions of such equations.

1997, 3(3): 317-332 doi: 10.3934/dcds.1997.3.317 +[Abstract](2224) +[PDF](221.3KB)
Abstract:
We prove the local simultaneous linearizability of a pair of commuting holomorphic functions at a shared fixed point under a very general - we conjecture optimal - diophantine condition. Let $f,g :\mathbb{C} \to \mathbb{C}$ with a common fixed point at the origin and suppose that $f(z) = \lambda z + \cdots$ and $\lambda \ne 0$. The map, $f,$ is called linearizable if there is an analytic diffeomorphism, $h$, which conjugates $f$ with its linear part in a neighborhood of the origin, i.e., $h^{-1} \circ f \circ h (z) = \lambda z$ where $\lambda = f'(0).$ Two such diffeomorphisms are simultaneously linearizable if they are linearized by the same map, $h$. If $|\lambda| = 1$ then the situation is delicate. Nonlinearizable maps are topologically abundant, i.e., for $\lambda$ in a dense co-meager set in $\mathbb{S}^1$ there exist nonlinearizable analytic maps with linear coefficient $\lambda$. In contrast there is a diophantine condition on $\lambda$ that is satisfied by a set of full measure in $\mathbb{S}^1$ which assures linearizability of the map $f$.
1997, 3(3): 333-340 doi: 10.3934/dcds.1997.3.333 +[Abstract](2266) +[PDF](181.7KB)
Abstract:
In this paper, we study the elliptic boundary value problem in a bounded domain $\Omega$ in $R^n$, with smooth boundary:

$-\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$

$u(x) = 0 \quad \quad x \in \partial \Omega.$

where $R(x)$ is a smooth function that may change signs. In [2], using a blowing up argument, Berestycki, Dolcetta, and Nirenberg, obtained a priori estimates and hence the existence of solutions for the problem when the exponent $1 < p < {n+2}/{n-1}$. Inspired by their result, in this article, we use the method of moving planes to fill the gap between ${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.

1997, 3(3): 341-370 doi: 10.3934/dcds.1997.3.341 +[Abstract](2651) +[PDF](339.5KB)
Abstract:
We consider optimal control problems governed by semilinear par- abolic equations with nonlinear boundary conditions and pointwise constraints on the state variable. In Robin boundary conditions considered here, the nonlinear term is neither necessarily monotone nor Lipschitz with respect to the state variable. We derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces. The adjoint state must satisfy a parabolic equation with Radon measures in Robin boundary conditions, in the terminal condition and in the distributed term. We give a precise meaning to the adjoint equation with measures as data and we prove the existence of a unique weak solution for this equation in an appropriate space.
1997, 3(3): 371-382 doi: 10.3934/dcds.1997.3.371 +[Abstract](2448) +[PDF](194.2KB)
Abstract:
This paper deals with a problem, when the invariant hyperbolic tori for Hamiltonian systems persist under perturbation. An existence theorem about such invariant tori is proved. Because the unperturbed systems possess the stronger degeneracy, this generalizes the classical KAM theorem and a well known result of Graff.
1997, 3(3): 383-400 doi: 10.3934/dcds.1997.3.383 +[Abstract](2713) +[PDF](224.6KB)
Abstract:
We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation

$i u_t + u_{x x} + ia(|u|^2u)_x = 0, \quad (t,x) \in \mathbf{R}\times \mathbf{R},$

$u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS)

where $a \in \mathbf{R}$. We prove that if $||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$ is sufficiently small with $\gamma > 1/2$, then the solution of (DNLS) satisfies the time decay estimate

$||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2},$

where $H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}= ||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2} < \infty\}$, $m,s\in \mathbf{R}$. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that $\gamma \ge 2$. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].

1997, 3(3): 401-418 doi: 10.3934/dcds.1997.3.401 +[Abstract](1884) +[PDF](282.6KB)
Abstract:
We study the existence of periodic solutions for the infinite-dimensional second order system $\ddot x=V_{x},\ x\in\mathbb{T}^{\mathbb{Z}_+}.$ Using the Implicit-Function-Theorem, we prove the existence of time-periodic solutions at "high frequencies"; no "smallness condition" on $V(x)$ is required.
1997, 3(3): 419-432 doi: 10.3934/dcds.1997.3.419 +[Abstract](1953) +[PDF](239.9KB)
Abstract:
We investigate how the non-analytic solitary wave solutions -- peakons and compactons -- of an integrable bi-Hamiltonian system arising in fluid mechanics, can be recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for the reduced dynamical system. This phenomenon is examined to understand the important effect of linear dispersion terms on the analyticity of such homoclinic orbits.
1997, 3(3): 433-438 doi: 10.3934/dcds.1997.3.433 +[Abstract](2302) +[PDF](161.1KB)
Abstract:
The logarithmic expansion rate of a positively invariant set for a $C^1$ endomorphism is shown to equal the infimum of the Lyapunov exponents for ergodic measures with support in the invariant set. Using this result, aperiodic flows of the two torus are shown to have an expansion rate of zero and the effects of conjugacies on expansion rates are investigated.
1997, 3(3): 439-450 doi: 10.3934/dcds.1997.3.439 +[Abstract](2140) +[PDF](207.5KB)
Abstract:
We look for $T$-periodic solutions on a convex Riemannian manifold $\mathcal{M}$ of the differential equation

$D_s\dot x(s) + \nabla V_x(x(s),s) = 0$

where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.

1997, 3(3): 451-455 doi: 10.3934/dcds.1997.3.451 +[Abstract](2106) +[PDF](156.0KB)
Abstract:
We prove the existence of periodic solutions for perturbations of some autonomous second order nonlinear differential equations by the use of the Poincaré- Birkhoff fixed point theorem.

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