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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 1

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Analysis of the Laplacian and spectral operators on the Vicsek set
Sarah Constantin , Robert S. Strichartz and  Miles Wheeler
2011, 10(1): 1-44 doi: 10.3934/cpaa.2011.10.1 +[Abstract](51) +[PDF](2669.8KB)
Abstract:
We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann boundary conditions. We study gaps in the ratios of eigenvalues and eigenvalue clusters. We give an explicit formula for the Green's function on $\mathcal{VS}_n$. Finally, we explain how the spectrum of the Laplacian on $\mathcal{VS}_n$ converges as $n \to \infty$ to the spectrum of the Laplacian on two crossed lines (the limit of the sets $\mathcal{VS}_n$.)
The existence of weak solutions for a generalized Camassa-Holm equation
Shaoyong Lai , Qichang Xie , Yunxi Guo and  YongHong Wu
2011, 10(1): 45-57 doi: 10.3934/cpaa.2011.10.45 +[Abstract](45) +[PDF](384.6KB)
Abstract:
A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.
The obstacle problem for Monge-Ampère type equations in non-convex domains
Jingang Xiong and  Jiguang Bao
2011, 10(1): 59-68 doi: 10.3934/cpaa.2011.10.59 +[Abstract](51) +[PDF](360.4KB)
Abstract:
In this paper, we consider the obstacle problem for Monge-Ampère type equations which include prescribed Gauss curvature equation as a special case. We establish $C^{1,1}$ regularity of the greatest viscosity solution in non-convex domains.
A comparison principle for a Sobolev gradient semi-flow
Timothy Blass , Rafael De La Llave and  Enrico Valdinoci
2011, 10(1): 69-91 doi: 10.3934/cpaa.2011.10.69 +[Abstract](34) +[PDF](511.7KB)
Abstract:
We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.
    We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.
    We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.
The heterogeneous dam problem with leaky boundary condition
Samia Challal and  Abdeslem Lyaghfouri
2011, 10(1): 93-125 doi: 10.3934/cpaa.2011.10.93 +[Abstract](35) +[PDF](485.0KB)
Abstract:
We study the heterogeneous dam problem, assuming the ux at the bottoms of the reservoirs obeying to a nonlinear law called leaky boundary condition. The velocity and the pressure are related by a nonlinear Darcy's law. Under a general monotonicity hypothesis on the permeability matrix, we prove that the free boundary is represented locally by graphs of continuous functions. We also prove the uniqueness of minimal and maximal solutions. When the ow is given by a linear Darcy law and the permeability matrix is symmetric, we prove the uniqueness of the reservoirs-connected solution.
Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation
Benjamin Dodson
2011, 10(1): 127-140 doi: 10.3934/cpaa.2011.10.127 +[Abstract](91) +[PDF](397.6KB)
Abstract:
We prove global well-posedness for the cubic, defocusing, nonlinear Schrödinger equation on $R^2$ with data $u_0 \in H^s(R^2)$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].
Traveling waves and their stability in a coupled reaction diffusion system
Xiaojie Hou and  Wei Feng
2011, 10(1): 141-160 doi: 10.3934/cpaa.2011.10.141 +[Abstract](56) +[PDF](459.6KB)
Abstract:
We study the traveling wave solutions to a reaction diffusion system modeling the public goods game with altruistic behaviors. The existence of the waves is derived through monotone iteration of a pair of classical upper- and lower solutions. The waves are shown to be unique and strictly monotonic. A similar KPP wave like asymptotic behaviors are obtained by comparison principle and exponential dichotomy. The stability of the traveling waves with non-critical speed is investigated by spectral analysis in the weighted Banach spaces.
Blowing up at zero points of potential for an initial boundary value problem
Jong-Shenq Guo and  Masahiko Shimojo
2011, 10(1): 161-177 doi: 10.3934/cpaa.2011.10.161 +[Abstract](52) +[PDF](443.2KB)
Abstract:
We study nonnegative radially symmetric solutions for a semilinear heat equation in a ball with spatially dependent coefficient which vanishes at the origin. Our aim is to construct a solution that blows up at the origin where there is no reaction. For this, we first prove that the blow-up is complete, if the origin is not a blow-up point and if there is no blow-up point on the boundary. Then we prove that a threshold solution exists such that it blows up in finite time incompletely and there is no blow-up point on the boundary. On the other hand, we prove that any zero of nonnegative potential is not a blow-up point for a more general problem under the assumption that the solution is monotone in time.
Boundedness in a class of duffing equations with oscillating potentials via the twist theorem
Huiping Jin
2011, 10(1): 179-192 doi: 10.3934/cpaa.2011.10.179 +[Abstract](48) +[PDF](421.2KB)
Abstract:
In this paper, we prove the boundedness of all solutions and the existence of periodic and quasi-periodic solutions for the equation $\ddot{x}+x^{2n+1}+\sum_{j=0}^l x^j p_j (x,t)=0$, where $p_j (x,t)$ are smooth 1-periodic functions in both $x$ and $t$ with $n\geq 1, 0 \leq l \leq 2 n$.
Asymptotic behavior for solutions of some integral equations
Yutian Lei and  Chao Ma
2011, 10(1): 193-207 doi: 10.3934/cpaa.2011.10.193 +[Abstract](94) +[PDF](386.7KB)
Abstract:
In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$

$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,

$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $

We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.

Asymptotic behavior of solutions to a model system of a radiating gas
Yongqin Liu and  Shuichi Kawashima
2011, 10(1): 209-223 doi: 10.3934/cpaa.2011.10.209 +[Abstract](36) +[PDF](418.1KB)
Abstract:
In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates of solutions. Moreover, we show that the solution is asymptotic to the linear diffusion wave which is given in terms of the heat kernel.
Principal curvature estimates for the level sets of harmonic functions and minimal graphs in $R^3$
Xi-Nan Ma , Jiang Ye and  Yun-Hua Ye
2011, 10(1): 225-243 doi: 10.3934/cpaa.2011.10.225 +[Abstract](35) +[PDF](367.5KB)
Abstract:
We give a sharp lower bound for the principal curvature of the level sets of harmonic functions and minimal graphs defined on convex rings in $R^3$ with homogeneous Dirichlet boundary conditions.
Singular positive solutions for a fourth order elliptic problem in $R$
Tokushi Sato and  Tatsuya Watanabe
2011, 10(1): 245-268 doi: 10.3934/cpaa.2011.10.245 +[Abstract](36) +[PDF](465.5KB)
Abstract:
In this paper, we consider the following fourth order elliptic problem in $R^N$:

$\Delta^2 u-c_1\Delta u+c_2 u=u^p+\kappa \sum_{i=1}^m \alpha_i \delta_{a_i}$ in $\mathcal D'(R^N),$

$ u(x)>0, u(x) \rightarrow 0 $ as $ |x| \rightarrow \infty. $

We will prove if $0 < \kappa < \kappa^* $ for some $\kappa^*\in (0,\infty)$, then this problem has at least two singular positive solutions.

Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials
Jun Wang , Junxiang Xu and  Fubao Zhang
2011, 10(1): 269-286 doi: 10.3934/cpaa.2011.10.269 +[Abstract](111) +[PDF](458.8KB)
Abstract:
In this paper we study the following nonperiodic second order Hamiltonian system

$-\ddot{u}(t)+L(t)u(t)=\nabla_u R(t,u(t)), \forall (t,u)\in R\times R^N, $

where the matrix $L(t)\in C(R,R^{N^2})$ and $R(t,u)$ is asymptotically quadratic or super quadratic in $u$ as $|u|\rightarrow\infty$. Under more general assumptions on the matrix $L(t)$, if $R$ is superquadratic and even in $u$, we obtain infinitely many homoclinic orbits. On the other hand, if $R$ is asymptotically quadratic, we also prove the existence and multiplicity of homoclinic orbits for the above system.

Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces
Zhichun Zhai
2011, 10(1): 287-308 doi: 10.3934/cpaa.2011.10.287 +[Abstract](51) +[PDF](501.3KB)
Abstract:
In this paper, we study two types of generalized Keller-Segel system of chemotaxis. We establish the global existence and uniqueness of solutions to the semilinear Keller-Segel system of doubly parabolic type and the nonlinear nonlocal type Keller-Segel system with data in Besov spaces. Moreover, we prove the stability of solution to the first type. Our main tools are the $L^p-L^q$ estimates for $e^{-t(-\triangle)^{\theta/2}}$ in Besov spaces and the perturbation of linearization.
Regularity criteria for a magnetohydrodynamic-$\alpha$ model
Yong Zhou and  Jishan Fan
2011, 10(1): 309-326 doi: 10.3934/cpaa.2011.10.309 +[Abstract](41) +[PDF](397.6KB)
Abstract:
We study the $n$-dimensional magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model in the whole space. Various regularity criteria are established. When $n=4$, uniqueness of weak solutions is also proved. As a corollary, the strong solution to this model exists globally, as $n \leq 4$.
Critical points of solutions to elliptic problems in planar domains
Jaime Arango and  Adriana Gómez
2011, 10(1): 327-338 doi: 10.3934/cpaa.2011.10.327 +[Abstract](34) +[PDF](871.9KB)
Abstract:
Given a planar domain $\Omega$, and an analytic function $f$, we describe the set of critical points for the solution $u$ of the semilinear elliptic problem $\Delta u = f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$. For simply connected domains we establish that the set of critical points is finite while for non--simply connected domains we show that this set is made up of finitely many isolated points and finitely many analytic Jordan curves. Further results are given in the case that $\Omega$ is an annular domain whose border has nonzero curvature.
On the global attractor of the Trojan Y Chromosome model
Rana D. Parshad and  Juan B. Gutierrez
2011, 10(1): 339-359 doi: 10.3934/cpaa.2011.10.339 +[Abstract](46) +[PDF](562.7KB)
Abstract:
We consider the Trojan Y Chromosome (TYC) model for eradication of invasive species in population dynamics. We present global estimates for the TYC system in a spatial domain. In this work we prove the existence of a global attractor for the system. We derive uniform estimates to tackle the question of asymptotic compactness of the semi-group for the TYC model in $H^2(\Omega)$. This along with the existence of a bounded absorbing set, which we also derive, demonstrates the existence of a global attractor for the TYC model. The present analysis reveals that extinction of an invasive species is always possible to achieve irrespective of geometric considerations of the domain. This result is valid for TYC systems in which advection is negligible. This theoretical work lays the foundation for experimental studies of the application of the TYC eradication strategy in spatial ecology, since the outcome is in principle guaranteed.
An evolution equation involving the normalized $P$-Laplacian
Kerstin Does
2011, 10(1): 361-396 doi: 10.3934/cpaa.2011.10.361 +[Abstract](82) +[PDF](1438.7KB)
Abstract:
This paper considers an initial-boundary value problem for the evolution equation associated with the normalized $p$-Laplacian. There exists a unique viscosity solution $u,$ which is globally Lipschitz continuous with respect to $t$ and locally with respect to $x.$ Moreover, we study the long time behavior of the viscosity solution $u$ and compute numerical solutions of the problem.

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