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Communications on Pure & Applied Analysis

2006 , Volume 5 , Issue 1

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Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities
V. Mastropietro and Michela Procesi
2006, 5(1): 1-28 doi: 10.3934/cpaa.2006.5.1 +[Abstract](267) +[PDF](271.5KB)
Abstract:
We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichelet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by a convergent perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.
Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions
B. Abdellaoui, E. Colorado and I. Peral
2006, 5(1): 29-54 doi: 10.3934/cpaa.2006.5.29 +[Abstract](245) +[PDF](269.3KB)
Abstract:
We study the following parabolic problem

$u_t-$ div $(|x|^{-p\gamma}|\nabla u|^{p-2}\nabla u) = \lambda f(x,u), u\ge 0$ in $\Omega\times (0,T)$,

$ B(u) = 0$ on $\partial\Omega\times (0,T),$

$ u(x,0) = \varphi (x)\quad$ if $x\in\Omega$,

where $\Omega\subset\mathbb R^N$ is a smooth bounded domain with $0\in\Omega$,

$B(u)\equiv u\chi_{\Sigma_1\times(0,T)}+|x|^{-p\gamma} |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\chi_{\Sigma_2 \times (0,T)}$

and $-\infty<\gamma<\frac{N-p}{p}$. The boundary conditions over $\partial\Omega\times (0,T)$ verify hypotheses that will be precised in each case.
Mainly, we will consider the second member $f(x,u)=\frac{u^{\alpha}}{|x|^{p(\gamma+1)}}$ with $ \alpha\ge p-1$, as a model case. The main points under analysis are some existence, nonexistence and complete blow-up results related to some Hardy-Sobolev inequalities and a weak version of Harnack inequality, that holds for $p\ge 2$ and $\gamma+1>0$.

Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems
Ahmed Y. Abdallah
2006, 5(1): 55-69 doi: 10.3934/cpaa.2006.5.55 +[Abstract](207) +[PDF](176.3KB)
Abstract:
We investigate the existence of the global attractor and its upper semicontinuity for the lattice dynamical system of a Klein-Gordon-Schrödinger type equation in a suitable Hilbert space.
On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian
C. Cortázar and Marta García-Huidobro
2006, 5(1): 71-84 doi: 10.3934/cpaa.2006.5.71 +[Abstract](195) +[PDF](161.9KB)
Abstract:
We consider the problem of uniqueness of radial ground state solutions to

$(P)\qquad\qquad\qquad\qquad -\Delta u=K(|x|)f(u),\quad x\in \mathbb R^n. $

Here $K$ is a positive $C^1$ function defined in $\mathbb R^+$ and $f\in C[0,\infty)$ has one zero at $u_0>0$, is non positive and not identically 0 in $(0,u_0)$, and it is locally lipschitz, positive and satisfies some superlinear growth assumption in $(u_0,\infty)$.

Spike solutions to a nonlocal differential equation
Changfeng Gui and Zhenbu Zhang
2006, 5(1): 85-95 doi: 10.3934/cpaa.2006.5.85 +[Abstract](256) +[PDF](140.1KB)
Abstract:
In this paper we consider a nonlocal differential equation, which is a limiting equation of one dimensional Gierer-Meinhardt model. We study the existence of spike steady states and their stability. We also construct a single-spike quasi-equilibrium solution and investigate the dynamics of spike-like solutions.
The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system
Masaki Kurokiba, Toshitaka Nagai and T. Ogawa
2006, 5(1): 97-106 doi: 10.3934/cpaa.2006.5.97 +[Abstract](216) +[PDF](132.6KB)
Abstract:
In this paper, we discuss the global existence and uniform boundedness of the radial solutions to the drift-diffusion system in two space dimension, which is derived from the simulation of semiconductor device design and self-interacting particles. It is shown that the time global existence and the uniform boundedness of the solution to the problem below the sharp threshold condition.
On the strong invariance property for non-Lipschitz dynamics
Mikhail Krastanov, Michael Malisoff and Peter Wolenski
2006, 5(1): 107-124 doi: 10.3934/cpaa.2006.5.107 +[Abstract](242) +[PDF](233.8KB)
Abstract:
We provide a new sufficient condition for strong invariance for differential inclusions, under very general conditions on the dynamics, in terms of a Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the multifunction, we assume a feedback realization condition that can in particular be satisfied for measurable dynamics that are neither upper nor lower semicontinuous.
A Nekhoroshev theorem for some infinite--dimensional systems
Paolo Perfetti
2006, 5(1): 125-146 doi: 10.3934/cpaa.2006.5.125 +[Abstract](241) +[PDF](330.6KB)
Abstract:
We study the persistence for long times of the solutions of some infinite--dimensional discrete hamiltonian systems with formal hamiltonian $\sum_{i=1}^\infty h(A_i) + V(\varphi),$ $(A,\varphi)\in \mathbb R^{\mathbb N}\times \mathbb T^{\mathbb N}.$ $V(\varphi)$ is not needed small and the problem is perturbative being the kinetic energy unbounded. All the initial data $(A_i(0), \varphi_i(0)),$ $i\in \mathbb N$ in the phase--space $\mathbb R^{\mathbb N} \times \mathbb T^{\mathbb N},$ give rise to solutions with $|A_i(t) - A_i(0)|$ close to zero for exponentially--long times provided that $A_i(0)$ is large enough for $|i|$ large. We need $\frac{\partial h}{\partial A_i}(A_i(0))$ unbounded for $i\to+\infty$ making $\varphi_i$ a fast variable the greater is $i,$ the faster is the angle $\varphi_i$ (avoiding the resonances). The estimates are obtained in the spirit of the averaging theory reminding the analytic part of Nekhoroshev--theorem.
Boundary layer and variational eigencurve in two-parameter single pendulum type equations
Tetsutaro Shibata
2006, 5(1): 147-154 doi: 10.3934/cpaa.2006.5.147 +[Abstract](226) +[PDF](116.0KB)
Abstract:
We consider the nonlinear two-parameter single pendulum type equation $-u''(t) + \mu f(u(t)) = \lambda\sin u(t), t \in I$ :$= (-T, T), u(t) > 0, t \in I, u(\pm T) = 0$, where $T > 0$ is a constant and $\mu, \lambda > 0$ are parameters. For a given $\mu > 0$, there exists a solution triple $(\mu, \lambda(\mu), u_\mu) \in \mbox{\bf R}_+^2 \times C^2(\bar{I})$, which is obtained by a variational method, such that $u_\mu$ develops a boundary layer as $\mu \to \infty$. We establish the precise asymptotic formulas for $||u_\mu||_\infty, u_\mu'(\pm T)$ and the variational eigencurve $\lambda(\mu)$ as $\mu \to \infty$.
Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients
Luisa Moschini, Guillermo Reyes and Alberto Tesei
2006, 5(1): 155-179 doi: 10.3934/cpaa.2006.5.155 +[Abstract](245) +[PDF](244.9KB)
Abstract:
We prove nonuniqueness of solutions of the Cauchy problem for a semilinear parabolic equation with inverse-square potential in certain Lebesgue spaces. The nonuniqueness results proved in [5] are the limiting case of the present ones as the strength of the potential vanishes. Similar results are obtained for a related semilinear parabolic equation with singular coefficients. The proofs rely on investigating by variational methods in suitable weighted Sobolev spaces the equation satisfied by the profile of a radial similarity solution.
Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration
C. M. Elliott, B. Gawron, S. Maier-Paape and E. S. Van Vleck
2006, 5(1): 181-200 doi: 10.3934/cpaa.2006.5.181 +[Abstract](163) +[PDF](688.6KB)
Abstract:
In this article we consider a model that generalizes the Perona-Malik and the total variation models. We consider discretizations of this new model and show that the discretizations conserve certain properties of the continuous model, in particular convergence of the iterative scheme to a critical point and existence of a discrete Liapunov functional. Computational results are obtained that illustrate different features of the family of models.
Controllability of couette flows
Michael Schmidt and Emmanuel Trélat
2006, 5(1): 201-211 doi: 10.3934/cpaa.2006.5.201 +[Abstract](221) +[PDF](257.6KB)
Abstract:
In this article, we investigate the problem of controlling Navier-Stokes equations between two infinite rotating coaxial cylinders. We prove that it is possible to move from a given Couette flow, that is a special stationary solution, to another one, by controlling the rotation velocity of the outer cylinder.
Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients
Pierpaolo Soravia
2006, 5(1): 213-240 doi: 10.3934/cpaa.2006.5.213 +[Abstract](278) +[PDF](263.8KB)
Abstract:
In this paper we prove the comparison principle for viscosity solutions of second order, degenerate elliptic pdes with a discontinuous, inhomogeneous term having discontinuities on Lipschitz surfaces. It is shown that appropriate sub and supersolutions $u,v$ of a Dirichlet type boundary value problem satisfy $u\leq v$ in $\Omega$. In particular, continuous viscosity solutions are unique. We also give examples of existence results and apply the comparison principle to prove convergence of approximations.

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