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Communications on Pure & Applied Analysis

2007 , Volume 6 , Issue 1

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Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle
Magdalena Caubergh, Freddy Dumortier and Robert Roussarie
2007, 6(1): 1-21 doi: 10.3934/cpaa.2007.6.1 +[Abstract](186) +[PDF](225.5KB)
Abstract:
It is known that perturbations from a Hamiltonian 2-saddle cycle $\Gamma $can produce limit cycles that are not covered by the Abelian integral, even when the Abelian integral is generic. These limit cycles are called alien limit cycles. In this paper, extending the results of [6] and [2], we investigate the number of alien limit cycles in generic multi-parameter rigid unfoldings of the Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation.
Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system
Francisco Ortegón Gallego and María Teresa González Montesinos
2007, 6(1): 23-42 doi: 10.3934/cpaa.2007.6.23 +[Abstract](186) +[PDF](206.7KB)
Abstract:
We show the existence of a capacity solution to a coupled nonlinear parabolic--elliptic system, the elliptic part in the parabolic equation being of the form -div $a(x,t,u,\nabla u)$, where the operator $a$ is of Leray--Lions type. Also, we consider the case where the elliptic equation is non-uniformly elliptic. The system may be regarded as a generalized version of the well-known thermistor problem.
Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian
Xavier Cabré and Manel Sanchón
2007, 6(1): 43-67 doi: 10.3934/cpaa.2007.6.43 +[Abstract](138) +[PDF](260.5KB)
Abstract:
We consider nonnegative solutions of $-\Delta_p u=f(x,u)$, where $p>1$ and $\Delta_p$ is the $p$-Laplace operator, in a smooth bounded domain of $\mathbb R^N$ with zero Dirichlet boundary conditions. We introduce the notion of semi-stability for a solution (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the properties of this class of solutions.
Under some assumptions on $f$ that make its growth comparable to $u^m$, we prove that every semi-stable solution is bounded if $m < m_{c s}$. Here, $m_{c s}=m_{c s}(N,p)$ is an explicit exponent which is optimal for the boundedness of semi-stable solutions. In particular, it is bigger than the critical Sobolev exponent $p^\star-1$.
We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and $1 < p < 2$.
The stability of the equilibrium for a perturbed asymmetric oscillator
Xiong Li
2007, 6(1): 69-82 doi: 10.3934/cpaa.2007.6.69 +[Abstract](153) +[PDF](161.8KB)
Abstract:
In this paper we will derive some stability criteria for the equilibrium of a perturbed asymmetric oscillator

$\ddot x +a^+ x^+ - a^-$ $x^-$ $+ b(t)x^2+r(t,x)=0,$

where $a^+,a^-$ are two different positive numbers, $b(t)$ is a $2\pi$-periodic function, and the remaining term $r(t,x)$ is $2\pi$-periodic with respect to the time $t$ and dominated by the power $x^3$ in a neighborhood of the equilibrium $x=0$.

Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory
Tamara Fastovska
2007, 6(1): 83-101 doi: 10.3934/cpaa.2007.6.83 +[Abstract](143) +[PDF](218.4KB)
Abstract:
A nonlinear problem for thermoelastic Mindlin-Timoshenko plate with hereditary heat conduction of Gurtin-Pipkin type is considered here. We prove the existence of a compact global attractor whose fractal dimension is finite. The main aim of the work is to show the upper semicontinuity of the attractor as the relaxation time tends to zero.
Periodic solutions of nonlinear periodic differential systems with a small parameter
Adriana Buică, Jean–Pierre Françoise and Jaume Llibre
2007, 6(1): 103-111 doi: 10.3934/cpaa.2007.6.103 +[Abstract](171) +[PDF](130.3KB)
Abstract:
We deal with nonlinear periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide sufficient conditions in order that some of the periodic orbits of this invariant manifold persist after the perturbation. These conditions are not difficult to check, as we show in some applications. The key tool for proving the main result is the Lyapunov--Schmidt reduction method applied to the Poincaré--Andronov mapping.
Global attractor for a composite system of nonlinear wave and plate equations
Francesca Bucci, Igor Chueshov and Irena Lasiecka
2007, 6(1): 113-140 doi: 10.3934/cpaa.2007.6.113 +[Abstract](207) +[PDF](270.4KB)
Abstract:
We prove the existence of a compact, finite dimensional, global attractor for a system of strongly coupled wave and plate equations with nonlinear dissipation and forces. This kind of models describes fluid-structure interactions. Though our main focus is on the composite system of two partial differential equations, the result achieved yields as well a new contribution to the asymptotic analysis of either (uncoupled) equation.
Some nonexistence results for quasilinear PDE's
Evgeny Galakhov
2007, 6(1): 141-161 doi: 10.3934/cpaa.2007.6.141 +[Abstract](201) +[PDF](216.3KB)
Abstract:
We establish nonexistence results for some quasilinear partial differential equations of elliptic, parabolis, and hyperbolic types using the nonlinear capacity method.
A variational approach to resonance for asymmetric oscillators
D. Bonheure and C. Fabry
2007, 6(1): 163-181 doi: 10.3934/cpaa.2007.6.163 +[Abstract](144) +[PDF](197.8KB)
Abstract:
We consider in this note the equation

$x'' + \alpha x^+ - \beta x^- + g(x) =p(t),$

where $x^+ =$ max{$x,0$} is the positive part of $x$, $x^- $ =max{$-x,0$} its negative part and $\alpha,\beta$ are positive parameters. We assume that $g :\mathbb R \to \mathbb R$ is continuous and bounded on $\mathbb R$, $p:\mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of $2\pi$-periodic solutions when $(\alpha,\beta)$ belongs to one of the curves of the Fučík spectrum corresponding to $2\pi$-periodic boundary conditions.

Localization of blow-up points for a nonlinear nonlocal porous medium equation
Lili Du and Zheng-An Yao
2007, 6(1): 183-190 doi: 10.3934/cpaa.2007.6.183 +[Abstract](179) +[PDF](130.8KB)
Abstract:
This paper deals with the porous medium equation with a nonlinear nonlocal source

$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$

subject to homogeneous Dirichlet condition. We investigate the influence of the nonlocal source and local term on blow-up properties for this system. It is proved that: (i) when $p\leq 1$, the nonlocal source plays a dominating role, i.e. the system has global blow-up and the blow-up profile which is uniformly away from the boundary either at polynomial scale or logarithmic scale is obtained. (ii) When $p > m$, this system presents single blow-up pattern. In other words, the local term dominates the nonlocal term in the blow-up profile. This extends the work of Li and Xie in Appl. Math. Letter, 16 (2003) 185--192.

A result on Hardy-Sobolev critical elliptic equations with boundary singularities
Jinhui Chen and Haitao Yang
2007, 6(1): 191-201 doi: 10.3934/cpaa.2007.6.191 +[Abstract](181) +[PDF](204.0KB)
Abstract:
In this note, a Hardy-Sobolev critical elliptic equation with boundary singularities and sublinear perturbation is studied. We obtain a result on the existence of classical solution and the multiplicity of weak solutions by making use of sub-super solutions and variational methods.
Remarks on dispersive estimates and curvature
Fabio Nicola
2007, 6(1): 203-212 doi: 10.3934/cpaa.2007.6.203 +[Abstract](142) +[PDF](152.1KB)
Abstract:
We investigate connections between certain dispersive estimates of a (pseudo) differential operator of real principal type and the number of non-vanishing curvatures of its characteristic manifold. More precisely, we obtain sharp thresholds for the range of Lebesgue exponents depending on the specific geometry.
Periodic solutions of a periodic scalar piecewise ode
José Luis Bravo, Manuel Fernández and Antonio Tineo
2007, 6(1): 213-228 doi: 10.3934/cpaa.2007.6.213 +[Abstract](140) +[PDF](189.7KB)
Abstract:
We study the distributions of periodic solutions of scalar piecewise equations defined by $x'=f(t,x)$ if $x\geq 0$, and $x'=g(t,x)$ if $x<0$, where $f,g$ are time periodic $\mathcal C^1$-functions such that $f(t,0)=g(t,0)$. Thus, these are equations on the cylinder where the vector field is not necessarily smooth on one of the equatorial circles.
We find that the solutions are $\mathcal C^1$-functions when the equation restricted to the equatorial line has a finite number of zeroes. Moreover, if $f$ and $g$ are analytic functions and the zeroes on the equatorial line are finite and simple, the set of periodic solutions consists of isolated periodic solutions and a finite number (determined by the number of zeroes) of closed "bands" of periodic solutions.
A note on the exact controllability for nonautonomous hyperbolic systems
Tatsien Li and Zhiqiang Wang
2007, 6(1): 229-235 doi: 10.3934/cpaa.2007.6.229 +[Abstract](151) +[PDF](142.9KB)
Abstract:
By choosing suitable examples, we show that, quite different from the autonomous hyperbolic case, the exact boundary controllability for nonautonomous hyperbolic systems possesses various possibilities.
Monotone solutions to a class of elliptic and diffusion equations
Li Ma, Chong Li and Lin Zhao
2007, 6(1): 237-246 doi: 10.3934/cpaa.2007.6.237 +[Abstract](186) +[PDF](135.9KB)
Abstract:
In this paper, we study the existence and properties of monotone solutions to the following elliptic equation in $\mathbf R^n$

$-\Delta u= F'(u),$ in $\mathbf R^n,$

$\partial_{x_n}u>0,$

and the diffusion equation

$u_t-\Delta u= F'(u),$ in $\mathbf R^n\times$ {$t>0$},

$\partial_{x_n}u>0, u|_{t=0}=u_0,$

where $\Delta$ is the standard Laplacian operator in $\mathbf R^n$, and $u_0$ is a given smooth function in $\mathbf R^n$ with some monotonicity condition. We show that under a natural condition on the nonlinear term $F'$, there exists a global solution to the diffusion problem above, and as time goes to infinity, the solution converges in $C_{l o c}^2(\mathbf R^n)$ to a solution to the corresponding elliptic problem. In particular, we show that for any two solutions $u_1(x')<$ $u_2(x')$ to the elliptic equation in $\mathbf R^{n-1}$:

$-\Delta u=F'(u),$ in $\mathbf R^{n-1}, $

either for every $c\in (u_1(0),u_2(0))$, there exists an $(n-1)$ dimensional solution $u_c$ with $u_c(0)=c$, or there exists an $x_n$-monotone solution $u(x',x_n)$ to the elliptic equation in $\mathbf R^n$:

$-\Delta u=F'(u), $ in $\mathbf R^n,$

$\partial_{x_n}u>0,$ in $\mathbf R^n$

such that

$\lim_{x_n\to-\infty}u(x',x_n)=v_1(x')\leq u_1(x')$

and

$\lim_{x_n\to+\infty}u(x',x_n)=v_2(x')\leq u_2(x').$

A typical example is when $F'(u)=u-|u|^{p-1}u$ with $p>1$. Some of our results are similar to results for minimizers obtained by Jerison and Monneau [13] by variational arguments. The novelty of our paper is that we only assume the condition for $F$ used by Keller and Osserman for boundary blow up solutions.

Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary
W. G. Litvinov
2007, 6(1): 247-277 doi: 10.3934/cpaa.2007.6.247 +[Abstract](164) +[PDF](267.8KB)
Abstract:
General conditions of slip of a fluid on the boundary are derived and a problem on stationary flow of the electrorheological fluid in which the terms of slip are specified on one part of the boundary and surface forces are given on the other is formulated and studied. Existence of a generalized (weak) solution of this problem is proved by using the methods of penalty functions, monotonicity and compactness. It is shown that the method of penalty functions and the Galerkin approximations can be used for the approximate solution of the problem under consideration. The existence and the uniqueness of the smooth classical solution of the problem is proved in the case that the conditions of slip are prescribed on the whole of the boundary.
Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows
Ting Li
2007, 6(1): 279-285 doi: 10.3934/cpaa.2007.6.279 +[Abstract](173) +[PDF](110.4KB)
Abstract:
In this paper we study the existence of pullback attractors for non-autonomous multi-valued semiflows. First, we prove abstract results on the existence of limit sets under the assumptions of pullback asymptotically upper semi-compact and pullback absorbing. Then, we prove the existence of pullback attractors. Further, we prove the existence of $D$-pullback attractor under weaker conditions.
Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology
F. R. Guarguaglini and R. Natalini
2007, 6(1): 287-309 doi: 10.3934/cpaa.2007.6.287 +[Abstract](137) +[PDF](239.7KB)
Abstract:
In this paper we establish general well-posedeness results for a wide class of weakly parabolic $2\times 2$ systems in a bounded domain of $\mathbb R^N$. Our results cover examples arising in sulphation of marbles and chemotaxis, when the density of one chemical component is not diffusing. We show that, under quite general assumptions, uniform $L^\infty$ estimates are sufficient to establish the global existence and stability of solutions, even if in general the nonlinear terms in the equations depend also on the gradient of the solutions. Applications are presented and discussed.

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