# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

December 2006 , Volume 5 , Issue 4

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2006, 5(4): 675-690 doi: 10.3934/cpaa.2006.5.675 +[Abstract](1332) +[PDF](175.4KB)
Abstract:
In this paper we study the optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential $V$ with respect to $V$, when the potential is restricted to a bounded, closed and convex set of $L^q(\Omega)$.
2006, 5(4): 691-708 doi: 10.3934/cpaa.2006.5.691 +[Abstract](1458) +[PDF](219.2KB)
Abstract:
The $L^2$-critical defocusing nonlinear Schrödinger initial value problem on $\mathbb R^d$ is known to be locally well-posed for initial data in $L^2$. Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data $u_0$ in Sobolev $H^1$ and for data in the weighted space $(1+|x|) u_0 \in L^2$. For the $d=2$ problem, it is known that global existence holds for data in $H^s$ and also for data in the weighted space $(1+|x|)^\sigma u_0 \in L^2$ for certain $s, \sigma < 1$. We prove: If global well-posedness holds in $H^s$ then global existence and scattering holds for initial data in the weighted space with $\sigma = s$.
2006, 5(4): 709-731 doi: 10.3934/cpaa.2006.5.709 +[Abstract](1713) +[PDF](230.3KB)
Abstract:
We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group.
2006, 5(4): 733-762 doi: 10.3934/cpaa.2006.5.733 +[Abstract](1412) +[PDF](255.5KB)
Abstract:
We prove existence of a renormalized solution to a system of nonlinear partial differential equations with anisotropic diffusivities and transport effects, supplemented with initial and Dirichlet boundary conditions. The data are assumed to be merely integrable. This system models the spread of an epidemic disease through a heterogeneous habitat.
2006, 5(4): 763-777 doi: 10.3934/cpaa.2006.5.763 +[Abstract](1193) +[PDF](168.3KB)
Abstract:
This note deals with a nonlinear system of PDEs accounting for phase transition phenomena. The existence of solutions of a related Cauchy-Neumann problem is established in the one-dimensional setting. A fixed point procedure guarantees the existence of solutions locally in time. Next, an argument based on a priori estimates allows to extend such solutions in the whole time interval. Hence, the uniqueness of the solution is proved by proper contracting estimates.
2006, 5(4): 779-792 doi: 10.3934/cpaa.2006.5.779 +[Abstract](1420) +[PDF](163.3KB)
Abstract:
The treatment by radiotherapy or chemotherapy to human cancers induces a complex chain of events involving reversible cell cycle and cell death [1]. In this paper we study the asymptotical behavior of the solutions for the mathematical model that has potential to describe the growth of human tumors cells and their responses to therapy. We found that the solutions of this system are either bounded, exponential bounded or exponential decay. This result can be used to predict the response of cells to mitotic arrest.
2006, 5(4): 793-812 doi: 10.3934/cpaa.2006.5.793 +[Abstract](1558) +[PDF](219.8KB)
Abstract:
The aim of this paper is twofold. We construct an extension to a non-integrable case of Hopf's formula, often used to produce viscosity solutions of Hamilton-Jacobi equations for $p$-convex integrable Hamiltonians. Furthermore, for a general class of $p$-convex Hamiltonians, we present a proof of the equivalence of the minimax solution with the viscosity solution.
2006, 5(4): 813-826 doi: 10.3934/cpaa.2006.5.813 +[Abstract](1438) +[PDF](160.6KB)
Abstract:
We consider the problem of uniqueness of radial ground state solutions to

(P) $\qquad\qquad\qquad -\Delta u=K(|x|)f(u),\quad x\in \mathbb R^n.$

Here $K$ is a positive $C^1$ function defined in $\mathbb R^+$ and $f\in C[0,\infty)$ has one zero at $u_0>0$, is non positive and not identically 0 in $(0,u_0)$, and it is locally lipschitz, positive and satisfies some superlinear growth assumption in $(u_0,\infty)$.

2006, 5(4): 827-838 doi: 10.3934/cpaa.2006.5.827 +[Abstract](1842) +[PDF](155.9KB)
Abstract:
A parabolic-hyperbolic nonconserved phase-field model is here analyzed. This is an evolution system consisting of a parabolic equation for the relative temperature $\theta$ which is nonlinearly coupled with a semilinear damped wave equation governing the order parameter $\chi$. The latter equation is characterized by a nonlinearity $\phi(\chi)$ with cubic growth. Assuming homogeneous Dirichlet and Neumann boundary conditions for $\theta$ and $\chi$, we prove that any weak solution has an $\omega$-limit set consisting of one point only. This is achieved by means of adapting a method based on the Łojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.
2006, 5(4): 839-853 doi: 10.3934/cpaa.2006.5.839 +[Abstract](1475) +[PDF](184.0KB)
Abstract:
In this paper, one--dimensional ($1D$) resonant beam equation

$u_{t t} +u_{x x x x} +u^3=0,$

with hinged boundary conditions is considered. It is proved that the above equation admits small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills.

2006, 5(4): 855-859 doi: 10.3934/cpaa.2006.5.855 +[Abstract](2376) +[PDF](102.3KB)
Abstract:
In this paper, we study a conjecture of J.Serrin and give a partial generalized result of the work of de Figueiredo and Felmer about Liouville type Theorem for non-negative solutions for an elliptic system. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality.
2006, 5(4): 861-885 doi: 10.3934/cpaa.2006.5.861 +[Abstract](1348) +[PDF](236.8KB)
Abstract:
In this paper we study controllability of control systems in $\mathbb R^n$ of the form $\dot x=f(x)+\sum_{i=1}^m$ $u_ig_i(x)$ with $u\in\mathcal U$ compact convex subset of $\mathbb R^n$ with a rather general target. The symmetric (driftless) case, i.e. $f=0$, is a very classical topic, and in this case the results on controllability and Hölder continuity of the minimal time function $T$ are related to certain properties of the Lie algebra generated by the $g_i$'s. Here, we want to extend some results on controllability and Hölder continuity of $T$ to some cases where $f\ne 0$.
2006, 5(4): 887-905 doi: 10.3934/cpaa.2006.5.887 +[Abstract](1345) +[PDF](228.9KB)
Abstract:
It is proved that the initial value problem for the fifth order Kadomtsev-Petviashvili (KPII) equation is locally well-posed in the anisotropic Sobolev spaces $H^{s_1,s_2}( \mathbb R^2)$ with $s_1$>$-\frac{5}{4}$ and $s_2\geq 0,$ and globally well-posed in $H^{s,0}(\mathbb R^2)$ with $s$>$-\frac{4}{7}.$
2006, 5(4): 907-918 doi: 10.3934/cpaa.2006.5.907 +[Abstract](1394) +[PDF](147.2KB)
Abstract:
In reference [13], by Constantin and Fefferman, a quite simple geometrical assumption on the direction of the vorticity is shown to be sufficient to guarantee the regularity of the weak solutions to the evolution Navier--Stokes equations in the whole of $\mathbf R^3$. Essentially, the solution is regular if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [8], among other side results, the authors prove that $1/2$-Hölder continuity is sufficient.
A main open problem remains of the possibility of extending the same kind of results to boundary value problems. Here, we succeed in making this extension to the well known Navier (or slip) boundary condition in the half-space $\mathbf R^3$. It is worth noting that the extension to the non-slip boundary condition remains open. See [7].
2006, 5(4): 919-928 doi: 10.3934/cpaa.2006.5.919 +[Abstract](1153) +[PDF](384.0KB)
Abstract:
This paper deals with the generalized Activator-Inhibitor model which originally arose in studies of pattern-formation in biology and has interesting and challenging mathematical properties. We study the long time existence of solutions as well as the boundedness and blowup properties for some special cases. We also obtain a priori estimates of stationary solutions followed by some numerical solutions with moving mesh methods.
2006, 5(4): 929-939 doi: 10.3934/cpaa.2006.5.929 +[Abstract](1160) +[PDF](160.1KB)
Abstract:
In this paper we study analytically a class of waves in the variant of the classical Boussinesq system given by

$\partial_t u = -\partial_x v - \alpha \partial_{x x x} v - \epsilon \partial_x(u v), \quad \partial_t v = - \partial_x u - \epsilon v \partial_x v,$

where $\epsilon$ is an small parameter and $\alpha \in (0,1)$. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for some values of $\alpha$, it contains solutions that are defined for large values of time and they are very close (of order $O(\epsilon)$) to a linear torus for long times (of order $O(\epsilon^{-1})$). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

2006, 5(4): 941-952 doi: 10.3934/cpaa.2006.5.941 +[Abstract](1934) +[PDF](168.7KB)
Abstract:
We study the existence of positive solutions of the singular quasilinear elliptic equation

-div$(|x|^{-a p}|\nabla u|^{p-2}\nabla u)=|x|^{-(a+1)p+c}f(x,u)$ in $\Omega$

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

where $p>1$. We use upper and lower--solutions methods, variational techniques and regularity theory.

2006, 5(4): 953-962 doi: 10.3934/cpaa.2006.5.953 +[Abstract](1807) +[PDF](126.9KB)
Abstract:
The approximate controllability and approximate null controllability of control systems governed by a class of abstract semilinear integrodifferential equations are discussed in this paper. Sufficient conditions for approximate controllability and approximate null controllability are obtained providing the approximate controllability and exact null controllability of the corresponding linear systems respectively. Finally, some examples are given to illustrate the application of the sufficient conditions.
2006, 5(4): 963-979 doi: 10.3934/cpaa.2006.5.963 +[Abstract](1309) +[PDF](209.9KB)
Abstract:
We analyze the limit as the speed of light $c\rightarrow\infty$ of the global entropy solutions of the $2\times 2$ relativistic Euler equations for the state $p=\kappa^2\rho^\gamma$ $( 1<\gamma<2 )$, and find that the limit is the entropy solution of the corresponding non-relativistic Euler equations.

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