# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

January 2010 , Volume 9 , Issue 1

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2010, 9(1): 1-21 doi: 10.3934/cpaa.2010.9.1 +[Abstract](2642) +[PDF](296.7KB)
Abstract:
Time-frequency methods are used to study a class of Fourier Integral Operators (FIOs) whose representation using Gabor frames is proved to be very efficient. Indeed, similarly to the case of shearlets and curvelets frames [10, 35], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty, 1}$ [33], for some Fourier multipliers [6] and metaplectic operators [14, 31]. Moreover, this paper provides the mathematical tools to numerically solving the Cauchy problem for Schr¨odinger equations using Gabor frames [17]. Finally, similar arguments can be employed to study other classes of FIOs [16].
2010, 9(1): 23-45 doi: 10.3934/cpaa.2010.9.23 +[Abstract](2610) +[PDF](287.5KB)
Abstract:
In this paper we analyze particular changes of variable, called time transformations, reducing a delay differential equation with a state-dependent delay to a delay differential equation with a prescribed non-state-dependent delay. We then employ these transformations to compute the breaking points of solutions and to derive optimal superconvergence results for Runge-Kutta methods for state-dependent equations.
2010, 9(1): 47-60 doi: 10.3934/cpaa.2010.9.47 +[Abstract](2116) +[PDF](223.7KB)
Abstract:
In this note we prove the existence of solutions for a class of first order non-linear hyperbolic systems. Given a possibly non conservative straight line system, we prove that all hyperbolic systems sufficiently near to the given one also admit global solutions. The total variation of the initial data is not assumed to be small. Stability estimates are also provided.
2010, 9(1): 61-75 doi: 10.3934/cpaa.2010.9.61 +[Abstract](1953) +[PDF](294.0KB)
Abstract:
Modelling a crystal with impurities we study an atomic chain of point masses with linear nearest neighbour interactions. We assume that the masses of the particles are normalised to 1, except for one heavy particle which has mass $M$. We investigate the macroscopic behaviour of such a system when $M$ is large, and time and space are scaled accordingly. As main result we derive a PDE for the light particles that is coupled with an ODE for the heavy particle.
2010, 9(1): 77-90 doi: 10.3934/cpaa.2010.9.77 +[Abstract](2589) +[PDF](1575.4KB)
Abstract:
In this paper, we study the nonlinear wave solutions of the generalized Camassa-Holm-Degasperis-Procesi equation $u_t-u_{x x t}+(1+b)u^2u_x=b u_x u_{x x}+u u_{x x x}$. Through phase analysis, several new types of the explicit nonlinear wave solutions are constructed. Our concrete results are: (i) For given $b> -1$, if the wave speed equals $\frac{1}{1+b}$, then the explicit expressions of the smooth solitary wave solution and the singular wave solution are given. (ii) For given $b> -1$, if the wave speed equals $1+b$, then the explicit expressions of the peakon wave solution and the singular wave solution are got. (iii) For given $b> -2$ and $b\ne -1$, if the wave speed equals $\frac{2+b}{2}$, then the explicit smooth solitary wave solution, the peakon wave solution and the singular wave solution are obtained. We also verify the correctness of these solutions by using the software Mathematica. Our work extends some previous results.
2010, 9(1): 91-102 doi: 10.3934/cpaa.2010.9.91 +[Abstract](2275) +[PDF](174.2KB)
Abstract:
In this article, we study the boundary controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval. The control acts on the first spatial derivative at the left endpoint. First, we prove that this control system is null controllable. It is done using a spectral analysis and the method of moments. Then, we introduce a boundary feedback law stabilizing to zero the solution of the closed-loop system.
2010, 9(1): 103-108 doi: 10.3934/cpaa.2010.9.103 +[Abstract](2048) +[PDF](119.8KB)
Abstract:
In this paper, we consider a (generalized) envelope of flows on compact metric spaces. This partly generalizes the notion of envelope of maps in discrete geometry ([3]). We clarify a certain distinction between the flow geometry and the discrete one, which is explained by showing that any !-limit set for an envelope of flows is an empty set, whereas it is nonempty in general in discrete case.
2010, 9(1): 109-140 doi: 10.3934/cpaa.2010.9.109 +[Abstract](2356) +[PDF](364.9KB)
Abstract:
We establish Hardy inequalities of the form

$\int_\Omega | \nabla u|_A^2 dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx, \qquad u \in H_0^1(\Omega) \qquad\qquad (1)$

where $E$ is a positive function defined in $\Omega$, -div$(A \nabla E)$ is a nonnegative nonzero finite measure in $\Omega$ which we denote by $\mu$ and where $A(x)$ is a $n \times n$ symmetric, uniformly positive definite matrix defined in $\Omega$ with $| \xi |_A^2:= A(x) \xi \cdot \xi$ for $\xi \in \mathbb{R}^n$. We show that (1) is optimal if $E=0$ on $\partial \Omega$ or $E=\infty$ on the support of $\mu$ and is not attained in either case. When $E=0$ on $\partial \Omega$ we show

$\int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx + \frac{1}{2} \int_\Omega \frac{u^2}{E} d \mu, \qquad u \in H_0^1(\Omega)\qquad (2)$

is optimal and not attained. Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for $p$≠ 2 which, in the case that $\mu$ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]).
We examine improved versions of the above inequalities of the form

$\int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$

Necessary and sufficient conditions on $V$ are obtained (in terms of the solvability of a linear pde) for (3) to hold. Analogous results involving improvements are obtained for the weighted versions.
In addition we obtain various results concerning the above inequalities valid for functions $u$ which are nonzero on the boundary of $\Omega$. We also examine the nonquadradic case ,ie. $p$ ≠2 of the above inequalities.

2010, 9(1): 141-160 doi: 10.3934/cpaa.2010.9.141 +[Abstract](2332) +[PDF](273.8KB)
Abstract:
In this paper we consider the unique solvability of initial-boundary value problems of semi-linear wave equations with space-time dependent coefficients and special mixed non-homogeneous boundary values which form the so-called boundary-like antiperiodic condition. The procedure in this paper is the combination of the Galerkin method and a contraction.
2010, 9(1): 161-192 doi: 10.3934/cpaa.2010.9.161 +[Abstract](2736) +[PDF](370.2KB)
Abstract:
System of partial differential equations with convolution terms and non-local nonlinearity describing oscillations of plate due to Berger's approach and with accounting for thermal regime in terms of Coleman-Gurtin and Gurtin-Pipkin law and fading memory of material is considered. The equation is transformed into a dynamical system in a suitable Hilbert space, which asymptotic behavior is analysed. Existence of a compact global attractor in this dynamical system and some of its properties are proved in this paper. Main tool in analysis of asymptotic behavior is stabilizability inequality.
2010, 9(1): 193-210 doi: 10.3934/cpaa.2010.9.193 +[Abstract](2005) +[PDF](239.2KB)
Abstract:
In this paper, we first prove the existence of compact kernel sections for dissipative non-autonomous Zakharov lattice dynamical system, then we obtain an upper bound of fractal dimension of the compact kernel sections, and finally we establish an upper semicontinuity of the compact kernel sections.
2010, 9(1): 211-231 doi: 10.3934/cpaa.2010.9.211 +[Abstract](2408) +[PDF](430.4KB)
Abstract:
We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is $2$, the exponent in the Weyl law is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and Weyl exponent $4/3$, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.
2010, 9(1): 233-248 doi: 10.3934/cpaa.2010.9.233 +[Abstract](2382) +[PDF](225.4KB)
Abstract:
In this paper, existence and multiplicity results for semilinear elliptic equations in whole space, with concave and convex nonlinearities and weight function which can change sign, are established. The study is based on variational methods, more precisely, minimization techniques and mountain pass Theorem without Palais-Smale condition.
2010, 9(1): 249-260 doi: 10.3934/cpaa.2010.9.249 +[Abstract](2003) +[PDF](168.9KB)
Abstract:
For the circular restricted 3-body and 4-Body problems in $\mathbb{R}^2$, we prove the existence of new symmetric noncollision periodic solutions with some fixed winding numbers and masses.

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