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Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 1

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Stability of planar switched systems: The nondiagonalizable case
Moussa Balde and Ugo Boscain
2008, 7(1): 1-21 doi: 10.3934/cpaa.2008.7.1 +[Abstract](139) +[PDF](265.1KB)
Abstract:
Consider the planar linear switched system $\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where $A$ and $B$ are two $2\times 2$ real matrices, $x\in \mathbb R^2$, and $u(.):[0,\infty[\to$ {$0,1$} is a measurable function. In this paper we consider the problem of finding a (coordinate-invariant) necessary and sufficient condition on $A$ and $B$ under which the system is asymptotically stable for arbitrary switching functions $u(.)$.
This problem was solved in previous works under the assumption that both $A$ and $B$ are diagonalizable. In this paper we conclude this study, by providing a necessary and sufficient condition for asymptotic stability in the case in which $A$ and/or $B$ are not diagonalizable.
To this purpose we build suitable normal forms for $A$ and $B$ containing coordinate invariant parameters. A necessary and sufficient condition is then found without looking for a common Lyapunov function but using "worst-trajectory" type arguments.
Bifurcation and stability of two-dimensional double-diffusive convection
Chun-Hsiung Hsia, Tian Ma and Shouhong Wang
2008, 7(1): 23-48 doi: 10.3934/cpaa.2008.7.23 +[Abstract](165) +[PDF](298.8KB)
Abstract:
In this article, we present a bifurcation and stability analysis on the double-diffusive convection. The main objective is to study 1) the mechanism of the saddle-node bifurcation and hysteresis for the problem, 2) the formation, stability and transitions of the typical convection structures, and 3) the stability of solutions. It is proved in particular that there are two different types of transitions: continuous and jump, which are determined explicitly using some physical relevant nondimensional parameters. It is also proved that the jump transition always leads to the existence of a saddle-node bifurcation and hysteresis phenomena.
On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes
Oleg Makarenkov and Paolo Nistri
2008, 7(1): 49-61 doi: 10.3934/cpaa.2008.7.49 +[Abstract](157) +[PDF](244.0KB)
Abstract:
We consider an autonomous system in $\mathbb R^n$ having a limit cycle $ x_0$ of period $T>0$ which is nondegenerate in a suitable sense, (see Definition 2.1). We then consider the perturbed system obtained by adding to the autonomous system a $T$-periodic, not necessarily differentiable, term whose amplitude tends to $0$ as a small parameter $\varepsilon>0$ tends to $0.$ Assuming the existence of a $T$-periodic solution $x_\varepsilon$ of the perturbed system and its convergence to $ x_0$ as $\varepsilon\to 0$, the paper establishes the existence of $\Delta_\varepsilon\to 0$ as $\varepsilon\to 0$ such that $||x_\varepsilon(t+\Delta_\varepsilon)-x_0(t)||\le\varepsilon M$ for some $M>0$ and any $\varepsilon>0$ sufficiently small. This paper completes the work initiated by the authors in [4] and [11]. Indeed, in [4] the existence of a family of $T$-periodic solutions $x_\varepsilon$ of the perturbed system considered here was proved. While in [11] for perturbed systems in $\mathbb R^2$ the rate of convergence was investigated by means of the method considered in this paper.
Potential well method for initial boundary value problem of the generalized double dispersion equations
Yacheng Liu and Runzhang Xu
2008, 7(1): 63-81 doi: 10.3934/cpaa.2008.7.63 +[Abstract](314) +[PDF](257.0KB)
Abstract:
In this paper we study the initial boundary value problem of the generalized double dispersion equations $u_{t t}-u_{x x}-u_{x x t t}+u_{x x x x}=f(u)_{x x}$, where $f(u)$ include convex function as a special case. By introducing a family of potential wells we first prove the invariance of some sets and vacuum isolating of solutions, then we obtain a threshold result of global existence and nonexistence of solutions. Finally we discuss the global existence of solutions for problem with critical initial condition.
A variational argument to finding global solutions of a quasilinear Schrödinger equation
Jianqing Chen
2008, 7(1): 83-88 doi: 10.3934/cpaa.2008.7.83 +[Abstract](177) +[PDF](133.0KB)
Abstract:
We prove a sharp existence result of global solutions of the quasilinear Schrödinger equation

$iu_t + u_{x x} + |u|^{p-2}u +(|u|^2)_{x x}u = 0,\quad u|_{t=0}=u_0(x),\quad x\in \mathbb R$

for a large class of initial data. The result gives a qualitative description on how small an initial data can ensure the existence of global solutions which sharpen a global existence result with small initial data [7, 10].

Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$
Abbas Moameni
2008, 7(1): 89-105 doi: 10.3934/cpaa.2008.7.89 +[Abstract](151) +[PDF](199.5KB)
Abstract:
We study the existence of positive solutions to the quasilinear elliptic problem

$-\epsilon \Delta u+V(x)u-\epsilon k(\Delta(|u|^2))u=g(u), \quad u>0,x \in \mathbb R^N,$

where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

Everywhere regularity for P-harmonic type systems under the subcritical growth
Shenzhou Zheng, Laping Zhang and Zhaosheng Feng
2008, 7(1): 107-117 doi: 10.3934/cpaa.2008.7.107 +[Abstract](181) +[PDF](168.2KB)
Abstract:
In the present paper, we show that the derivative of any weak solution of p-harmonic type system under the subcritical growth belongs to a local Hölder continuity space with certain Hölder exponent. This conclusion is sharp in the sense of lower-order item growth.
Refinement of the Benoist theorem on the size of Dini subdifferentials
Ludovic Rifford
2008, 7(1): 119-124 doi: 10.3934/cpaa.2008.7.119 +[Abstract](164) +[PDF](118.3KB)
Abstract:
Given a lower semicontinuous function $f:\mathbb R^n \rightarrow \mathbb R \cup$ {$+\infty$}, we prove that the set of points of $\mathbb R^n$ where the lower Dini subdifferential has convex dimension $k$ is countably $(n-k)$-rectifiable. In this way, we extend a theorem of Benoist(see [1, Theorem 3.3]), and as a corollary we obtain a classical result concerning the singular set of locally semiconcave functions.
Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains
Luca Rossi
2008, 7(1): 125-141 doi: 10.3934/cpaa.2008.7.125 +[Abstract](185) +[PDF](259.1KB)
Abstract:
In this paper we consider fully nonlinear elliptic operators of the form $F(x,u,Du,D^2u)$. Our aim is to prove that, under suitable assumptions on $F$, the only nonnegative viscosity super-solution $u$ of $F(x,u,Du,D^2u)=0$ in an unbounded domain $\Omega$ is $u\equiv 0$. We show that this uniqueness result holds for the class of nonnegative super-solutions $u$ satisfying

limin$f_{x\in\Omega, |x|\to\infty}\frac{u(x)+1}{\dist(x,\partial\Omega)}=0,$

and then, in particular, for strictly sublinear super-solutions in a domain $\Omega$ containing an open cone. In the special case that $\Omega=\mathbb R^N$, or that $F$ is the Bellman operator, we show that the same result holds for the whole class of nonnegative super-solutions.
Our principal assumption on the operator $F$ involves its zero and first order dependence when $|x|\to\infty$. The same kind of assumption was introduced in a recent paper in collaboration with H. Berestycki and F. Hamel [4] to establish a Liouville type result for semilinear equations. The strategy we follow to prove our main results is the same as in [4], even if here we consider fully nonlinear operators, possibly unbounded solutions and more general domains.

Doubling property of elliptic equations
Bo Su
2008, 7(1): 143-147 doi: 10.3934/cpaa.2008.7.143 +[Abstract](164) +[PDF](119.0KB)
Abstract:
Doubling property of elliptic equations with integrable coefficients is shown.
On a class of anisotropic nonlinear elliptic equations in $\mathbb R^N$
Francois van Heerden and Zhi-Qiang Wang
2008, 7(1): 149-162 doi: 10.3934/cpaa.2008.7.149 +[Abstract](128) +[PDF](212.2KB)
Abstract:
We establish some imbedding results of weighted Sobolev spaces. The results then are used to obtain ground state solutions of nonlinear elliptic equations with anisotropic coefficients.
Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms
Olivier Guibé and Anna Mercaldo
2008, 7(1): 163-192 doi: 10.3934/cpaa.2008.7.163 +[Abstract](161) +[PDF](341.6KB)
Abstract:
In the present paper we prove uniqueness results for weak solutions to a class of problems whose prototype is

$-d i v((1+|\nabla u|^2)^{(p-2)/2} \nabla u)-d i v(c(x) (1+|u|^2)^{(\tau+1)/2}) $

$+b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$, $p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients $c(x)$ and $b(x)$ belong to suitable Lebesgue spaces, $f$ is an element of the dual space $W^{-1,p'}(\Omega)$ and $\tau$ and $\sigma$ are positive constants which belong to suitable intervals specified in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation
Yue Liu
2008, 7(1): 193-209 doi: 10.3934/cpaa.2008.7.193 +[Abstract](182) +[PDF](186.8KB)
Abstract:
We establish a sharp instability theorem for the standing-wave solutions of the inhomogeneous nonlinear Schrödinger equation

$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $ p = 1 + 4/n, n \ge 2, $ under certain conditions on the inhomogeneous term $ V $ with a small $ \epsilon > 0. $ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.

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