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Communications on Pure & Applied Analysis

2006 , Volume 5 , Issue 3

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On the existance of minimizers of the variable exponent Dirichlet energy integral
Peter A. Hästö
2006, 5(3): 415-422 doi: 10.3934/cpaa.2006.5.415 +[Abstract](137) +[PDF](199.6KB)
In this note we consider the Dirichlet energy integral in the variable exponent case under minimal assumptions on the exponent. First we show that the Dirichlet energy integral always has a minimizer if the boundary values are in $L^\infty$. Second, we give an example which shows that if the so-called "jump-condition", known to be sufficient, is violated, then a minimizer need not exist for unbounded boundary values.
Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking
Zhicheng Wang and Jun Wu
2006, 5(3): 423-433 doi: 10.3934/cpaa.2006.5.423 +[Abstract](161) +[PDF](191.5KB)
Applying the generalized Mawhin's continuation theorem of coincidence degree, we obtain some easily verifiable conditions for the existence of the positive periodic solutions of the following system

$x_1'(t)=h_1(t,x_1(t))(a_1(t)-a_{1 1}(t)x_1(t)-\frac{a_{1 3}(t)x_3(t)}{m(t)x_3(t)+x_1(t)})+D_1(t)(x_2(t)-x_1(t))+S_1(t),$

$x_2'(t)=h_2(t,x_2(t))(a_2(t)-a_{2 2}(t)x_2(t))+D_2(t)(x_1(t)-x_2(t))+S_2(t),$

$x_3'(t)=h_3(t,x_3(t))(-a_3(t)+\frac{a_{3 1}(t)x_1(t-\tau)}{m(t)x_3(t-\tau)+x_1(t-\tau)})+S_3(t).$

Some corresponding results are generalized or improved.

Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system
Qiong Chen, Chunlai Mu and Zhaoyin Xiang
2006, 5(3): 435-446 doi: 10.3934/cpaa.2006.5.435 +[Abstract](267) +[PDF](203.3KB)
This paper deals with the blow-up properties and asymptotic behavior of solutions to a semilinear integrodifferential system with nonlocal reaction terms in space and time. The blow-up conditions are given by a variant of the eigenfunction method combined with new properties on systems of differential inequalities. At the same time, the blow-up set is obtained. For some special cases, the asymptotic behavior of the blow-up solution is precisely characterized.
Stable determination of a semilinear term in a parabolic equation
Mourad Choulli, El Maati Ouhabaz and Masahiro Yamamoto
2006, 5(3): 447-462 doi: 10.3934/cpaa.2006.5.447 +[Abstract](206) +[PDF](235.3KB)
In [5] the first and third authors establish Hölder type stability estimates for the inverse problem consisting in the determination of a semilinear term of a parabolic equation from a single boundary measurement when the domain is a rectangle. In the present paper we extend the results in [5] for a general smooth domain.
Multiple bubbling for the exponential nonlinearity in the slightly supercritical case
Manuel del Pino, Jean Dolbeault and Monica Musso
2006, 5(3): 463-482 doi: 10.3934/cpaa.2006.5.463 +[Abstract](190) +[PDF](267.8KB)
Let $B$ denote the unit ball in $\mathbb R^2$. We consider the slightly super-critical Gelfand problem for the $p$-Laplacian operator $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$,

$ -\Delta_{2-\varepsilon} u=\lambdae^u$ in $B\quad u =0 $ on $ \partial B,$

for small $\varepsilon>0$. We show that if $k\ge 1$ is given and $\lambda>0$ is fixed and small, then there is a family of radial solutions exhibiting multiple blow-up as $\varepsilon\to 0$ in the form of a superposition of $k$ bubbles of different blow-up orders and shapes. Similar phenomena is found for the same problem involving the operator $\Delta_{N-\varepsilon}$ in $\mathbb R^N$, $N\ge 3$.

The study on solutions to Camassa-Holm equation with weak dissipation
Danping Ding, Lixin Tian and Gang Xu
2006, 5(3): 483-492 doi: 10.3934/cpaa.2006.5.483 +[Abstract](241) +[PDF](152.0KB)
This paper investigates the existence of solution on the weak dissipative Camassa-Holm equation, and studies the dynamical behaviors of the solution for this equation. An interesting result is given that the solution for the Cauchy problem on weak dissipative Camassa-Holm equation is local under certain conditions. Moreover, discussions on solitary solution to weak dissipation Camassa-Holm equation are given.
Dangerous Border-Collision bifurcations of a piecewise-smooth map
Hun Ki Baek and Younghae Do
2006, 5(3): 493-503 doi: 10.3934/cpaa.2006.5.493 +[Abstract](184) +[PDF](636.0KB)
In this paper we study the dangerous border-collision bifurcations [8] which recently have been numerically found on piecewise smooth maps characterized by non-differentiability on some surface in the phase space. The striking feature of such bifurcations is characterized by exhibiting a stable fixed point before and after the critical bifurcation point, but the unbounded behavior of orbits at the critical bifurcation point. We consider a specific variable space in order to do an analytical investigation of such bifurcations and prove the stability of fixed points. We also extend these bifurcation phenomena for the fixed points to the multiple coexisting attractors.
A boundary point lemma for Black-Scholes type operators
Erik Ekström and Johan Tysk
2006, 5(3): 505-514 doi: 10.3934/cpaa.2006.5.505 +[Abstract](133) +[PDF](134.9KB)
We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.
The stability of the equilibrium for a perturbed asymmetric oscillator
Xiong Li
2006, 5(3): 515-528 doi: 10.3934/cpaa.2006.5.515 +[Abstract](159) +[PDF](214.6KB)
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 $x=0$.

Nonlocal Cauchy problems for nonautonomous evolution equations
Jin Liang, James H. Liu and Ti-Jun Xiao
2006, 5(3): 529-535 doi: 10.3934/cpaa.2006.5.529 +[Abstract](195) +[PDF](162.9KB)
This paper is concerned with nonlocal Cauchy problems for semilinear nonautonomous evolution equations with compact evolution families on Banach spaces. A new existence result about mild solutions for the above problems is obtained without Lipschitz conditions on nonlinear and nonlocal terms.
On a criterium of global attraction for discrete dynamical systems
B. Coll, A. Gasull and R. Prohens
2006, 5(3): 537-550 doi: 10.3934/cpaa.2006.5.537 +[Abstract](159) +[PDF](242.1KB)
Consider that the origin is a fix point of a discrete dynamical system $x^{(n+1)}=F(x^{(n)})$, defined in the whole $\mathbb R^m.$ LaSalle, in his book of 1976, [13], proposes to study several conditions which might imply global attraction. One of his suggestions is to write $F(x)=A(x)x$, where $A(x)$ is a real $m\times m$ matrix, and to assume that all the eigenvalues of eigenvalues of $A(x)$, for all $x\in \mathbb R^m$, have modulus smaller than one. In the paper [4], Cima et al. show that, when $m\ge2$, such hypothesis does not guarantee that the origin is a global attractor, even for polynomial maps $F$. From the observation that the decomposition of $F(x)$ as $A(x)x$ is not unique, in this paper we wonder whether LaSalle condition, for a special and canonical choice of $A,$ forces the origin to be a global attractor. This canonical choice is given by $A_c(x)=\int_0^1 DF(sx) ds,$ where the integration of the matrix $DF(x)$ is made term by term. In fact, we prove that LaSalle condition for $A_c(x)$ is a sufficient condition to get the global attraction of the origin when $m=1,$ or when $m=2$ and $F$ is polynomial. We also show that this is no more true for $m=2$ when $F$ is a rational map or when $m\ge3.$ Finally we consider the equivalent question for ordinary differential equations.
Input-output conditions for the asymptotic behavior of linear skew-product flows and applications
Bogdan Sasu and A. L. Sasu
2006, 5(3): 551-569 doi: 10.3934/cpaa.2006.5.551 +[Abstract](198) +[PDF](246.9KB)
In this paper we present a new approach concerning the uniform exponential dichotomy of linear skew-product flows and extend existing results on exponential dichotomy roughness for variational systems in infinite dimensional spaces. We introduce new concepts of admissibility and we deduce their connections with the uniform exponential dichotomy of discrete linear skew-product flows. We apply our results at the study of the exponential dichotomy roughness of discrete linear skew-product flows, presenting an estimation for the lower bound of the dichotomy radius.
Positive radial solutions for some quasilinear elliptic systems in exterior domains
João Marcos do Ó, Sebastián Lorca, Justino Sánchez and Pedro Ubilla
2006, 5(3): 571-581 doi: 10.3934/cpaa.2006.5.571 +[Abstract](196) +[PDF](210.4KB)
We use fixed-point theorem of cone expansion/compression type to prove the existence of positive radial solutions for the following class of quasilinear elliptic systems in exterior domains

$-\Delta_p u = k_1(|x| )f(u,v),$ for $|x| > 1$ and $x \in \mathbb R^N, $

$-\Delta_p v = k_2(|x|)g(u,v),$ for $|x| > 1 $ and $x \in \mathbb R^N, $

$u(x) = v(x) =0,$ for $|x| =1, $

$u(x), v(x) \rightarrow 0 $ as $|x| \rightarrow +\infty,$

where $1 < p < N $ and $\Delta_p u=$ div $(|\nabla u|^{p-2}\nabla u )$ is the p-Laplacian operator. We consider nonlinearities that are either superlinear or sublinear.

On a variational inequality for the Navier-Stokes operator with variable viscosity
G. M. de Araújo and S. B. de Menezes
2006, 5(3): 583-596 doi: 10.3934/cpaa.2006.5.583 +[Abstract](140) +[PDF](215.8KB)
In this paper we investigate the unilateral problem for the operator $L$ perturbed of Navier-Stokes operator in a cylindrical case, where

$Lu=u'-(\nu_0+\nu_1||u(t)||^2)\Delta u+(u.\nabla )u-f+\nabla p.$

The mixed problem for the operator $L$ was proposed by J. L. Lions [6]. Using an appropriate penalization, we obtain a variational inequality for the Navier-Stokes perturbed system.

A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation
Kazuhiro Kurata and Tatsuya Watanabe
2006, 5(3): 597-610 doi: 10.3934/cpaa.2006.5.597 +[Abstract](181) +[PDF](167.3KB)
In this paper, we are concerned with radially symmetric solutions with a vortex to a nonlinear Schrödinger equation:

-ħ$^2 \Delta v+($ ħ$^2\omega^2/|x|^2+V(x))v=f(v)$ in $\mathbf R^2.$

We give precise asymptotic profiles of solutions as ħ$\rightarrow 0$ by variational methods and ODE arguments.

A remark on the damped wave equation
V. Pata and Sergey Zelik
2006, 5(3): 611-616 doi: 10.3934/cpaa.2006.5.611 +[Abstract](201) +[PDF](152.8KB)
In this short note we present a direct method to establish the optimal regularity of the attractor for the semilinear damped wave equation with a nonlinearity of critical growth.
Young measure solutions of the two-dimensional Perona-Malik equation in image processing
Yan Chen and Kewei Zhang
2006, 5(3): 617-637 doi: 10.3934/cpaa.2006.5.617 +[Abstract](176) +[PDF](996.3KB)
For a given smooth initial value $u_0$, we construct sequences of approximate solutions $u_j$ in $W^{1,\infty}$ for the well-known Perona-Malik anisotropic diffusion model in image processing defined by $u_t-$ div $ [\rho(|\nabla u|^2)\nabla u]=0$ under the homogeneous Neumann condition, where $\rho(|X|^2)X=X/(1+|X|^2)$ for $X\in\mathbb R^2$. The Perona-Malik diffusion equation is of non-coercive forward-backward type. Our constructed approximate solutions satisfy the equation in the sense that $(u_j)_t-$ div$_x [\rho(|\nabla u_j|^2)\nabla u_j]\to 0$ strongly in $W^{-1,p}(Q_T)$ for all $1\leq p<\infty$, where $Q_T=(0,T)\times \Omega$ with $\Omega\subset\mathbb R^2$ the unit square. We also show, for any non-constant initial value $u_0$ that the approximate solutions $u_j$ do not converge to a solution, rather, they converge weakly to Young measure-valued solutions which can be represented partially explicitly. Our main idea is to convert the equation into a differential inclusion problem.
Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations
Enrique Fernández-Cara, Manuel González-Burgos and Luz de Teresa
2006, 5(3): 639-658 doi: 10.3934/cpaa.2006.5.639 +[Abstract](166) +[PDF](262.9KB)
This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder $\Omega \times (0,T)$. More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form $\omega \times (0,T)$, where $\omega \subset \Omega$. In the wave equation, the restriction of the solution to the heat equation to another set $\mathcal O \times (0,T)$ appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on $T$, $\omega$ and $\mathcal O$, the equations are simultaneously controllable.
Analysis of a variable time-step discretization for a phase transition model with micro-movements
Ulisse Stefanelli
2006, 5(3): 659-673 doi: 10.3934/cpaa.2006.5.659 +[Abstract](138) +[PDF](231.0KB)
This note deals with a semi-implicit time discretization with variable time-step of a phase transition model taking into account the microscopic movements of molecules. In particular, we focus on the study of an unconditionally stable and convergent approximation. Moreover, an a priori estimate for the discretization error is established.

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