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Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 2

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Discrete Schrödinger equations and dissipative dynamical systems
Mostafa Abounouh , H. Al Moatassime , J. P. Chehab , S. Dumont and  Olivier Goubet
2008, 7(2): 211-227 doi: 10.3934/cpaa.2008.7.211 +[Abstract](31) +[PDF](275.4KB)
We introduce a Crank-Nicolson scheme to study numerically the long-time behavior of solutions to a one dimensional damped forced nonlinear Schrödinger equation. We prove the existence of a smooth global attractor for these discretized equations. We also provide some numerical evidences of this asymptotical smoothing effect.
Integral and series representations of the dirac delta function
Y. T. Li and  R. Wong
2008, 7(2): 229-247 doi: 10.3934/cpaa.2008.7.229 +[Abstract](52) +[PDF](206.6KB)
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions; they also include series of products of Laguerre polynomials and of spherical harmonics. The methods used are essentially based on the asymptotic behavior of these special functions.
Dynamical behaviour of a large complex system
Jianfeng Feng , Mariya Shcherbina and  Brunello Tirozzi
2008, 7(2): 249-265 doi: 10.3934/cpaa.2008.7.249 +[Abstract](37) +[PDF](225.5KB)
Limit theorems for a linear dynamical system with random interactions are established. The theorems enable us to characterize the dynamics of a large complex system in details and assess whether a large complex system is weakly stable or unstable (see Definition 1 below).
Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions
Petru Jebelean
2008, 7(2): 267-275 doi: 10.3934/cpaa.2008.7.267 +[Abstract](26) +[PDF](149.7KB)
This paper deals with the existence of infinitely many solutions for the boundary value problem

$-( | u' | ^{p-2}u')' + \varepsilon |u|^{p-2}u= \nabla F(t,u), $ in $(0,T)$,

$((|u'|^{p-2}u')(0), $ $ -(|u'|^{p-2}u')(T))$ $\in \partial j(u(0), u(T)),$

where $\varepsilon \geq 0$, $p \in (1, \infty)$ are fixed, the convex function $j:\mathbb R^N \times \mathbb R^N \to (- \infty , +\infty ]$ is proper, even, lower semicontinuous and $F:(0,T) \times \mathbb R^N \to \mathbb R $ is a Carathéodory mapping, continuously differentiable and even with respect to the second variable.

Isentropic approximation of the steady Euler system in two space dimensions
Chong Liu and  Yongqian Zhang
2008, 7(2): 277-291 doi: 10.3934/cpaa.2008.7.277 +[Abstract](22) +[PDF](209.5KB)
On the assumption that the initial data are isentropic and of sufficiently small total variation, we can prove that the difference between the solutions of the steady full Euler system and steady isentropic Euler system with the same initial data can be bounded by the cube of the total variation of the initial perturbation.
Higher--order implicit function theorems and degenerate nonlinear boundary-value problems
Olga A. Brezhneva , Alexey A. Tret’yakov and  Jerrold E. Marsden
2008, 7(2): 293-315 doi: 10.3934/cpaa.2008.7.293 +[Abstract](50) +[PDF](294.2KB)
The first part of this paper considers the problem of solving an equation of the form $F(x, y)=0$, for $y = \varphi (x)$ as a function of $x$, where $F: X \times Y \rightarrow Z$ is a smooth nonlinear mapping between Banach spaces. The focus is on the case in which the mapping $F$ is degenerate at some point $(x^*, y^*)$ with respect to $y$, i.e., when $F'_y (x^*, y^*)$, the derivative of $F$ with respect to $y$, is not invertible and, hence, the classical Implicit Function Theorem is not applicable. We present $p$th-order generalizations of the Implicit Function Theorem for this case. The second part of the paper uses these $p$th-order implicit function theorems to derive sufficient conditions for the existence of a solution of degenerate nonlinear boundary-value problems for second-order ordinary differential equations in cases close to resonance. The last part of the paper presents a modified perturbation method for solving degenerate second-order boundary value problems with a small parameter.The results of this paper are based on the constructions of $p$-regularity theory, whose basic concepts and main results are given in the paper Factor--analysis of nonlinear mappings: $p$--regularity theory by Tret'yakov and Marsden (Communications on Pure and Applied Analysis, 2 (2003), 425--445).
Global attractors for a three-dimensional conserved phase-field system with memory
Gianluca Mola
2008, 7(2): 317-353 doi: 10.3934/cpaa.2008.7.317 +[Abstract](30) +[PDF](415.4KB)
We consider a conserved phase-field system on a tridimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature $\vartheta$. These effects are represented through a convolution integral whose relaxation kernel $k$ is a summable and decreasing function. Therefore the system consists of a linear integrodifferential equation for $\vartheta$ which is coupled with a viscous Cahn-Hilliard type equation governing the order parameter $\chi$. The latter equation contains a nonmonotone nonlinearity $\phi$ and the viscosity effects are taken into account by the term $-\alpha \Delta\chi_t$, for some $\alpha \geq 0$. Thus, we formulate a Cauchy-Neumann problem depending on $\alpha $. Assuming suitable conditions on $k$, we prove that this problem generates a dissipative strongly continuous semigroup $S^\alpha (t)$ on an appropriate phase space accounting for the past histories of $\vartheta$ as well as for the conservation of the spatial means of the enthalpy $\vartheta+\chi$ and of the order parameter. We first show, for any $\alpha \geq 0$, the existence of the global attractor $\mathcal A_\alpha $. Also, in the viscous case ($\alpha > 0$), we prove the finiteness of the fractal dimension and the smoothness of $\mathcal A_\alpha $.
Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents
F. R. Pereira
2008, 7(2): 355-372 doi: 10.3934/cpaa.2008.7.355 +[Abstract](28) +[PDF](245.4KB)
In this work we study the existence of multiple solutions for the non-homogeneous system

$ - \Delta U = AU + (u^p_+, v^p_+)+ F$ in $\Omega$

$ U = 0 $ on $ \partial\Omega,$

where $\Omega\subset \mathbb R^{N}$ is a bounded smooth domain; $U=(u,v), p=2^\star -1$, with $2^\star=\frac{2N}{N-2}, N \geq 3$; ${w_+}=$ max{ $w,0$} and $F \in L^s(\Omega)\times L^s(\Omega)$ for some $s>N$.
Using variational methods, we prove the existence of at least two solutions. The first is obtained explicitly by a direct calculation and the second via the Mountain Pass Theorem for the case $0< \mu_1 \leq \mu_2< \lambda_1$ or Linking Theorem if $\lambda_k < \mu_1 \leq \mu_2 < \lambda_{k+1}$, where $\mu_1, \mu_2$ are eigenvalues of symmetric matrix $A$ and $\lambda_j$ are eigenvalues of $(-\Delta, H_0^1(\Omega))$.

One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries
Xulong Qin , Zheng-An Yao and  Hongxing Zhao
2008, 7(2): 373-381 doi: 10.3934/cpaa.2008.7.373 +[Abstract](140) +[PDF](153.3KB)
A free-boundary problem is studied for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity that decreases (to zero) with decreasing density, i.e., $\mu=A\rho^\theta$, where $A$ and $\theta$ are positive constants. The existence and uniqueness of the global weak solutions are obtained with $\theta\in (0,1]$, which improves the previous results and no vacuum is developed in the solutions in a finite time provided the initial data does not contain vacuum.
On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function
Tsung-Fang Wu
2008, 7(2): 383-405 doi: 10.3934/cpaa.2008.7.383 +[Abstract](37) +[PDF](250.7KB)
In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result to prove that the semilinear elliptic equation with a sign-changing weight function has at least two positive solutions.
Regularity theory in Orlicz spaces for the poisson and heat equations
Huilian Jia , Lihe Wang , Fengping Yao and  Shulin Zhou
2008, 7(2): 407-416 doi: 10.3934/cpaa.2008.7.407 +[Abstract](68) +[PDF](167.0KB)
In this paper we study the regularity theory in Orlicz spaces for the Poisson and heat equations.
Estimates for the life-span of the solutions for some semilinear wave equations
Meng-Rong Li
2008, 7(2): 417-432 doi: 10.3934/cpaa.2008.7.417 +[Abstract](31) +[PDF](190.6KB)
In this paper we prove main result on blow-up rates, blow-up constants and some estimates for life-spans of the solutions for some initial-boundary value problems for semi-linear wave equations. Under some conditions the life-span $T\star$ can be estimated by

$\beta (k,\alpha)$: $=$ min{ $2^{3/2+1/2\alpha}\cdot\delta( k,\alpha )a(0) a'(0)^{-1}:k\in (0,1)$},

where $a(0)=\int_\Omegau_{0}(x)^{2}dx,$ $a'(0)=2\int_\Omega u_{0}( x) u_1(x) dx$ and $\delta(k,\alpha )$ is given by

$\delta(k,\alpha)$ :$=\frac{1}{k}(\frac{k^2}{1-k^2})^{\frac{\alpha }{1+2\alpha}}$ $(1-(1+(\frac{1}{ k^2}-1)^{\frac{\alpha}{1+2\alpha}})^{\frac{-1}{2\alpha} }).

Finding invariant tori with Poincare's map
Hsuan-Wen Su
2008, 7(2): 433-443 doi: 10.3934/cpaa.2008.7.433 +[Abstract](27) +[PDF](168.0KB)
We consider the existence problem of invariant tori for quasi-periodic equation. We regard quasi-periodic functions with $n$ frequencies as periodic functions of functions with $n-1$ frequencies, which constitute a function space. Then we define Poincare's return map of a given semiflow on the space whose fixed point corresponds to an invariant torus of the semiflow.
Perturbation from symmetry and multiplicity of solutions for elliptic problems with subcritical exponential growth in $\mathbb{R} ^2$
Cristina Tarsi
2008, 7(2): 445-456 doi: 10.3934/cpaa.2008.7.445 +[Abstract](32) +[PDF](181.5KB)
We consider the following boundary value problem

$ -\Delta u= g(x,u) + f(x,u)\quad x\in \Omega $

$u=0\quad x\in \partial \Omega$

where $g(x,-\xi )=-g(x,\xi)$ and $g$ has subcritical exponential growth in $\mathbb R^2$. Using the method developed by Bolle, we prove that this problem has infinitely many solutions under suitable conditions on the growth of $g(u)$ and $f(u)$.

Discrete-time theorems for the dichotomy of one-parameter semigroups
Ciprian Preda
2008, 7(2): 457-463 doi: 10.3934/cpaa.2008.7.457 +[Abstract](32) +[PDF](118.6KB)
Discrete-time sufficient conditions for the dichotomy of $C_0$-semigroups are obtained in the general case when it is not required that the kernel of the dichotomic projector to be $T(t)$-invariant. Thus are extended known results due to Datko, Pazy, Zabczyk.
Elias M. Guio and  Ricardo Sa Earp
2008, 7(2): 465-465 doi: 10.3934/cpaa.2008.7.465 +[Abstract](21) +[PDF](81.5KB)
Referring to our paper Existence and non-existence for a mean curvature equation in hyperbolic space published in this journal, 4 (2005), 549-568, the assumptions are missing in the Statements: Theorem 3.1 and Theorem 3.2 ( cf. p. 552, lines 3-6). In the Statement of height estimates (Theorem 3.1 and Theorem 3.2), the assumptions on the prescribed mean curvature $H(x)$ are: $|H(x)|\leqs 1$ or $|H(x)|=a$ (constant). In the Statement of the main existence result (Theorem 3.3) the assumptions on the prescribed mean curvature $H(x)$ are the same: $|H(x)|\leqs 1$ or $|H(x)|=a$ (constant).

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