# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

All Issues

## Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 5

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2008, 7(5): 1017-1047 doi: 10.3934/cpaa.2008.7.1017 +[Abstract](422) +[PDF](388.3KB)
Abstract:
Multicomponent reactive flows arise in many physical applications in sciences and engineering. The objective of this work is to develop a rigorous mathematical theory based on the principles of continuum mechanics.
2008, 7(5): 1049-1056 doi: 10.3934/cpaa.2008.7.1049 +[Abstract](410) +[PDF](146.2KB)
Abstract:
In this work we study the hidden regularity for solutions of mixed problem associated to the Kirchhoff model for small deformations of a membrane

$u''-M(||u||^2) \Delta u=0,$

when the initial data are considered in spaces with few regularity.

2008, 7(5): 1057-1075 doi: 10.3934/cpaa.2008.7.1057 +[Abstract](438) +[PDF](256.6KB)
Abstract:
In this paper we study the nonlinear elliptic problem involving nearly critical exponent $(P_\varepsilon ): -\Delta u=$ $|u|^{4/(n-2)+\varepsilon}u$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb R^n$, $n \geq 3$ and $\varepsilon$ is a positive real parameter. We show that, for $\varepsilon$ small, $(P_\varepsilon)$ has no sign-changing solutions with low energy which blow up at two points. Moreover, we prove that there is no sign-changing solutions which blow up at three points. We also show that $(P_\varepsilon)$ has no bubble-tower sign-changing solutions.
2008, 7(5): 1077-1090 doi: 10.3934/cpaa.2008.7.1077 +[Abstract](428) +[PDF](218.1KB)
Abstract:
This paper considers systems of balance law with a dissipative non local source. A global in time well posedness result is obtained. Estimates on the dependence of solutions from the flow and from the source term are also provided. The technique relies on a recent result on quasidifferential equations in metric spaces.
2008, 7(5): 1091-1107 doi: 10.3934/cpaa.2008.7.1091 +[Abstract](411) +[PDF](225.8KB)
Abstract:
We consider the following problem

$\Delta u=\lambda [\frac{1}{u^p}-\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p-1}{q-1})^{-1/(p-q)} ]$ on $\partial B$, $0 < u < \kappa$

in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.

2008, 7(5): 1109-1122 doi: 10.3934/cpaa.2008.7.1109 +[Abstract](325) +[PDF](198.3KB)
Abstract:
In this paper, we consider the existence of solutions for a critical elliptic system. We show that the problem possesses at least a high energy positive solution in non-contractible domains.
2008, 7(5): 1123-1143 doi: 10.3934/cpaa.2008.7.1123 +[Abstract](621) +[PDF](258.2KB)
Abstract:
We prove the local well-posedness of a 1-D quadratic nonlinear Schrödinger equation

$iu_t+u_{x x}=\bar u^2$

in $H^s(\mathbb R)$ for $s\ge -1$ and ill-posedness below $H^{-1}$. The same result for another quadratic nonlinearity $u^2$ was given by I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006), but the function space of solutions depended heavily on the special property of the nonlinearity $u^2$. We construct the solution space suitable for the nonlinearity $\bar u^2$.

2008, 7(5): 1145-1178 doi: 10.3934/cpaa.2008.7.1145 +[Abstract](388) +[PDF](383.8KB)
Abstract:
In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\mathbb Z^d$, to a much wider class of infinite graphs. In particular, concerning the subcritical regime, we show that the connectivity functions are analytic and decay exponentially in any bounded degree graph. In the supercritical phase, we are able to prove the analyticity of finite connectivity functions in a smaller class of graphs, namely, bounded degree graphs with the so called minimal cut-set property and satisfying a (very mild) isoperimetric inequality. On the other hand we show that the large distances decay of finite connectivity in the supercritical regime can be polynomially slow depending on the topological structure of the graph. Analogous analyticity results are obtained for the pressure of the Random Cluster Model on an infinite graph, but with the further assumptions of amenability and quasi-transitivity of the graph.
2008, 7(5): 1179-1191 doi: 10.3934/cpaa.2008.7.1179 +[Abstract](365) +[PDF](213.5KB)
Abstract:
We prove the persistence of the regularity in the Besov norm spaces for the solutions of the subcritical Quasi-Geostrophic Equations with small size initial data in $\dot B^{-(2\alpha-1),\infty}_\infty$.
2008, 7(5): 1193-1201 doi: 10.3934/cpaa.2008.7.1193 +[Abstract](478) +[PDF](161.1KB)
Abstract:
We prove that among all doubly connected domains of $\mathbb R^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch [12] in dimension 2, and by Harrell, Kröger and Kurata [10] and Kesavan [13] in any dimension.
We also prove that the same result remains valid when the ambient space $\mathbb R^n$ is replaced by the standard sphere $\mathbb S^n$ or the hyperbolic space $\mathbb H^n$.
2008, 7(5): 1203-1210 doi: 10.3934/cpaa.2008.7.1203 +[Abstract](384) +[PDF](145.3KB)
Abstract:
This paper is concerned with a one dimensional transport equation with a non-local velocity and critical diffusion. In [Ann. of Math. 162 (2005), 1377--1389], A. Córdoba, D. Córdoba and Marco A. Fontelos showed the finite time singularities for a family of initial data without diffusion, global existence with subcritical diffusion and the global existence for small initial data with critical diffusion. We prove the global existence with critical diffusion, removing the smallness constraint on the initial data.
2008, 7(5): 1211-1223 doi: 10.3934/cpaa.2008.7.1211 +[Abstract](400) +[PDF](174.3KB)
Abstract:
In this paper we discuss a variational method of constructing an action minimizing stochastic invariant measure for positive definite Lagrangian systems. Then we study some main properties of the stochastic minimal measures. Finally we give the definitions of stochastic Mather's functions with respect to the stochastic differential equation d$x=v(t)$d$t+\sigma(x)$d$w$ and prove their differentiability.
2008, 7(5): 1225-1235 doi: 10.3934/cpaa.2008.7.1225 +[Abstract](542) +[PDF](154.5KB)
Abstract:
This paper is concerned with the existence of positive periodic solutions to a nonlinear fourth-order differential equation. By virtue of the first positive eigenvalue of the linear equation corresponding to the nonlinear fourth order equation, we establish the existence result by using the fixed point index theory in a cone.
2008, 7(5): 1237-1253 doi: 10.3934/cpaa.2008.7.1237 +[Abstract](456) +[PDF](221.4KB)
Abstract:
This paper examines the bifurcation and stability of the solutions of the complex Ginzburg--Landau equation(CGLE). The structure of the bifurcated solutions shall be explored as well. We investigate two different modes of the CGLE. The first mode of the CGLE contains only an unstable cubic term and the second mode contains not only a cubic term but a quintic term. The solutions of the cubic CGLE bifurcate from the trivial solution to an attractor supercritically in some parameter range. However, for the cubic-quintic CGLE, a subcritical bifurcation is obtained. Due to the global attractor, we obtain a saddle node bifurcation point $\lambda_c$. By thoroughly investigating the structure and transition of the solutions of the CGLE, we confirm that the bifurcated solutions are homeomorphic to $S^1$ and contain steady state solutions.
2008, 7(5): 1255-1267 doi: 10.3934/cpaa.2008.7.1255 +[Abstract](425) +[PDF](401.7KB)
Abstract:
In this paper we study a model for a species confined in a bounded region. This species diffuses slowly, follows a logistic law in the habitat and there is a flux of population across the boundary of the habitat.
Basically, we give some theoretical results of the model depending on some parameters which appear in the model.
2008, 7(5): 1269-1273 doi: 10.3934/cpaa.2008.7.1269 +[Abstract](365) +[PDF](114.4KB)
Abstract:
Considering the positive solution of the following nonlinear elliptic Neumann problem

$\Delta_0 u-\lambda u+f(u)=0, u>0,\$ in $\Omega,\quad \frac{\partial u}{\partial\nu}=0\$ on $\partial\Omega$

where $\Omega$ is convex and $f(u)$ defined by (2). We prove that for $1< p_i < 5$, $i=1,\cdots, K$ and $\lambda$ small, the only solution to the above problem is constant. This can be seen as a generalization of Theorem 1 in [7].

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