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Communications on Pure & Applied Analysis

2013 , Volume 12 , Issue 1

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Laplacians on a family of quadratic Julia sets II
Tarik Aougab , Stella Chuyue Dong and  Robert S. Strichartz
2013, 12(1): 1-58 doi: 10.3934/cpaa.2013.12.1 +[Abstract](42) +[PDF](5441.4KB)
Abstract:
This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and the Basilica-in-Rabbit. We also study the spectra of Laplacians on covering spaces and infinite blowups of the Julia sets. In particular, for a generic infinite blowups there is pure point spectrum, while for periodic covering spaces the spectrum is a mixture of discrete and continuous parts.
Whitney-type extensions in quasi-metric spaces
Ryan Alvarado , Irina Mitrea and  Marius Mitrea
2013, 12(1): 59-88 doi: 10.3934/cpaa.2013.12.59 +[Abstract](38) +[PDF](611.5KB)
Abstract:
We discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J. McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H. Whitney in the Euclidean setting. These generalizations are quantitatively sharp.
Blowup in higher dimensional two species chemotactic systems
Piotr Biler , Elio E. Espejo and  Ignacio Guerra
2013, 12(1): 89-98 doi: 10.3934/cpaa.2013.12.89 +[Abstract](37) +[PDF](353.8KB)
Abstract:
This paper deals with blowup properties of solutions to multicomponent parabolic-elliptic Keller--Segel model of chemotaxis in higher dimensions.
Ground state solutions for quasilinear stationary Schrödinger equations with critical growth
Marco A. S. Souto and  Sérgio H. M. Soares
2013, 12(1): 99-116 doi: 10.3934/cpaa.2013.12.99 +[Abstract](29) +[PDF](422.6KB)
Abstract:
We establish the existence of ground state solution for quasilinear Schrödinger equations involving critical growth. The method used here is minimizing the gradient integral norm in a manifold defined by integrals involving the primitive of the nonlinearity function.
A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component
Zujin Zhang
2013, 12(1): 117-124 doi: 10.3934/cpaa.2013.12.117 +[Abstract](59) +[PDF](346.0KB)
Abstract:
We study the Cauchy problem for the 3D Navier-Stokes equations, and prove some scalaring-invariant regularity criteria involving only one velocity component.
A refined result on sign changing solutions for a critical elliptic problem
Yuxin Ge , Monica Musso , A. Pistoia and  Daniel Pollack
2013, 12(1): 125-155 doi: 10.3934/cpaa.2013.12.125 +[Abstract](39) +[PDF](547.8KB)
Abstract:
In this work, we consider sign changing solutions to the critical elliptic problem $\Delta u + |u|^{\frac{4}{N-2}}u = 0$ in $\Omega_\varepsilon$ and $u=0$ on $\partial\Omega_\varepsilon$, where $\Omega_\varepsilon:=\Omega-\left(\bigcup_{i=1}^m (a_i+\varepsilon\Omega_i)\right)$ for small parameter $\varepsilon>0$ is a perforated domain, $\Omega$ and $\Omega_i$ with $0\in \Omega_i$ ($\forall i=1,\cdots,m$) are bounded regular general domains without symmetry in $\mathbb{R}^N$ and $a_i$ are points in $\Omega$ for all $i=1,\cdots,m$. As $\varepsilon$ goes to zero, we construct by gluing method solutions with multiple blow up at each point $a_i$ for all $i=1,\cdots,m$.
Existence theory for a Poisson-Nernst-Planck model of electrophoresis
L. Bedin and  Mark Thompson
2013, 12(1): 157-206 doi: 10.3934/cpaa.2013.12.157 +[Abstract](36) +[PDF](712.0KB)
Abstract:
A system modeling the electrophoretic motion of a charged rigid macromolecule immersed in a incompressible ionized fluid is considered. The ionic concentration is governing by the Nernst-Planck equation coupled with the Poisson equation for the electrostatic potential, Navier-Stokes and Newtonian equations for the fluid and the macromolecule dynamics, respectively. A local in time existence result for suitable weak solutions is established, following the approach of [15].
On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations
Martino Bardi and  Gabriele Terrone
2013, 12(1): 207-236 doi: 10.3934/cpaa.2013.12.207 +[Abstract](36) +[PDF](548.1KB)
Abstract:
We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.
The point-wise estimates of solutions for semi-linear dissipative wave equation
Yongqin Liu
2013, 12(1): 237-252 doi: 10.3934/cpaa.2013.12.237 +[Abstract](29) +[PDF](418.4KB)
Abstract:
In this paper we focus on the global-in-time existence and the point-wise estimates of solutions to the initial value problem for the semi-linear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the point-wise decay estimates of solutions to the problem.
Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1
Anna Mercaldo , Julio D. Rossi , Sergio Segura de León and  Cristina Trombetti
2013, 12(1): 253-267 doi: 10.3934/cpaa.2013.12.253 +[Abstract](44) +[PDF](408.3KB)
Abstract:
We consider the solution $u_p$ to the Neumann problem for the $p$--Laplacian equation with the normal component of the flux across the boundary given by $g\in L^\infty(\partial\Omega)$. We study the behaviour of $u_p$ as $p$ goes to $1$ showing that they converge to a measurable function $u$ and the gradients $|\nabla u_p|^{p-2}\nabla u_p$ converge to a vector field $z$. We prove that $z$ is bounded and that the properties of $u$ depend on the size of $g$ measured in a suitable norm: if $g$ is small enough, then $u$ is a function of bounded variation (it vanishes on the whole domain, when $g$ is very small) while if $g$ is large enough, then $u$ takes the value $\infty$ on a set of positive measure. We also prove that in the first case, $u$ is a solution to a limit problem that involves the $1-$Laplacian. Finally, explicit examples are shown.
Gradient blowup solutions of a semilinear parabolic equation with exponential source
Zhengce Zhang and  Yanyan Li
2013, 12(1): 269-280 doi: 10.3934/cpaa.2013.12.269 +[Abstract](37) +[PDF](384.3KB)
Abstract:
In this paper, we consider the N-dimensional semilinear parabolic equation $ u_t=\Delta u+e^{|\nabla u|}$, for which the spatial derivative of solutions becomes unbounded in finite (or infinite) time while the solutions themselves remain bounded. We establish estimates of blowup rate as well as lower and upper bounds for the radial solutions. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.
On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence
Tomás Caraballo and  David Cheban
2013, 12(1): 281-302 doi: 10.3934/cpaa.2013.12.281 +[Abstract](37) +[PDF](491.8KB)
Abstract:
The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so--called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper [6] to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations).
Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains
Jun Yang and  Xiaolin Yang
2013, 12(1): 303-340 doi: 10.3934/cpaa.2013.12.303 +[Abstract](48) +[PDF](588.7KB)
Abstract:
For a singularly perturbed equation of inhomogeneous Allen-Cahn type with positive potential function in high dimensional general domain, we prove the existence of solutions, at least for some sequence of the positive parameter, which have clustered phase transition layers with mass centered close to a smooth closed stationary and non-degenerate hypersurface. Moreover, the interaction between neighboring layers is governed by a type of Jacobi-Toda system.
Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows
Jihong Zhao , Qiao Liu and  Shangbin Cui
2013, 12(1): 341-357 doi: 10.3934/cpaa.2013.12.341 +[Abstract](31) +[PDF](439.8KB)
Abstract:
In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solution. In order to figure out the relation between the solution obtained here and weak solutions of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.
Spectral method for deriving multivariate Poisson summation formulae
Gusein Sh. Guseinov
2013, 12(1): 359-373 doi: 10.3934/cpaa.2013.12.359 +[Abstract](26) +[PDF](375.3KB)
Abstract:
We show that using spectral theory of a finite family of pairwise commuting Laplace operators and the spectral properties of the periodic Laplace operator some analogues of the classical multivariate Poisson summation formula can be derived.
Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction
Belkacem Said-Houari and  Flávio A. Falcão Nascimento
2013, 12(1): 375-403 doi: 10.3934/cpaa.2013.12.375 +[Abstract](51) +[PDF](505.0KB)
Abstract:
The goal of this work is to study a model of the viscoelastic wave equation with nonlinear boundary/interior sources and a nonlinear interior damping. First, applying the Faedo-Galerkin approximations combined with the compactness method to obtain existence of regular global solutions to an auxiliary problem with globally Lipschitz source terms and with initial data in the potential well. It is important to emphasize that it is not possible to consider density arguments to pass from regular to weak solutions if one considers regular solutions of our problem where the source terms are locally Lipschitz functions. To overcome this difficulty, we use an approximation method involving truncated sources and adapting the ideas in [13] to show that the existence of weak solutions can still be obtained for our problem. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term, then the solution ceases to exist and blows up in finite time provided that the initial data are large enough.
A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth
Federica Sani
2013, 12(1): 405-428 doi: 10.3934/cpaa.2013.12.405 +[Abstract](40) +[PDF](473.1KB)
Abstract:
In this paper we give sufficient conditions for the existence of solutions of a biharmonic equation of the form

$ \Delta^2 u + V(x)u = f(u)$ in $\mathbb{R}^4$

where $V$ is a continuous positive potential bounded away from zero and the nonlinearity $f(s)$ behaves like $e^{\alpha_0 s^2}$ at infinity for some $\alpha_0>0$.
In order to overcome the lack of compactness due to the unboundedness of the domain $\mathbb{R}^4$, we require some additional assumptions on $V$. In the case when the potential $V$ is large at infinity we obtain the existence of a nontrivial solution, while requiring the potential $V$ to be spherically symmetric we obtain the existence of a nontrivial radial solution. In both cases, the main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$.

Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$
Minbo Yang and  Yanheng Ding
2013, 12(1): 429-449 doi: 10.3934/cpaa.2013.12.429 +[Abstract](48) +[PDF](450.1KB)
Abstract:
In this paper we consider the following modified version of nonlinear Schrödinger equation:

$-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=g(x,u) $

in $\mathbb{R}^N$, $N\geq 3$ and $g(x,u)$ is a superlinear but subcritical function. Applying variational methods we show the existence and multiplicity of solutions provided $\varepsilon$ is sufficiently small enough.

Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros
Xiaohui Yu
2013, 12(1): 451-459 doi: 10.3934/cpaa.2013.12.451 +[Abstract](44) +[PDF](370.4KB)
Abstract:
We study the multiplicity solutions for the nonlinear elliptic equation

$ -\mathcal{M}_{\lambda,\Lambda}^+ (D^2u)=f(u) $ in $\Omega$,

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

and a more general fully nonlinear elliptic equation

$ F(D^2u)=f(u) $ in $\Omega$,

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

where $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 3$, $f$ is a locally Lipschitz continuous function with superlinear growth at infinity. We will show that the equation has at least two positive solutions under some assumptions.

Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$
Sergey Zelik and  Jon Pennant
2013, 12(1): 461-480 doi: 10.3934/cpaa.2013.12.461 +[Abstract](43) +[PDF](486.2KB)
Abstract:
We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary polynomial growth and for the case of sufficiently strong singular potentials. For these cases, the uniqueness and further regularity of the obtained solutions are proved as well. We discuss also the analogous problems for the case of the so-called Cahn-Hilliard-Oono equation where, in addition, the dissipativity of the associated solution semigroup is established.
The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response
Shanshan Chen , Junping Shi and  Junjie Wei
2013, 12(1): 481-501 doi: 10.3934/cpaa.2013.12.481 +[Abstract](37) +[PDF](554.7KB)
Abstract:
A delayed diffusive predator-prey system with Holling type-II predator functional response subject to Neumann boundary conditions is considered here. The stability/instability of nonnegative equilibria and associated Hopf bifurcation are investigated by analyzing the characteristic equations. By the theory of normal form and center manifold, an explicit formula for determining the stability and direction of periodic solution bifurcating from Hopf bifurcation is derived.
Incompressible type euler as scaling limit of compressible Euler-Maxwell equations
Jianwei Yang , Ruxu Lian and  Shu Wang
2013, 12(1): 503-518 doi: 10.3934/cpaa.2013.12.503 +[Abstract](39) +[PDF](441.9KB)
Abstract:
In this paper, we study the convergence of time-dependent Euler-Maxwell equations to incompressible type Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the local existence of smooth solutions to the limit equations is proved by an iterative scheme. Moreover, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and the symmetric hyperbolic property of the systems.
Numerical study of a family of dissipative KdV equations
Jean-Paul Chehab and  Georges Sadaka
2013, 12(1): 519-546 doi: 10.3934/cpaa.2013.12.519 +[Abstract](33) +[PDF](1743.3KB)
Abstract:
The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization e ects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization e ect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).
Resolvent estimates for a two-dimensional non-self-adjoint operator
Wen Deng
2013, 12(1): 547-596 doi: 10.3934/cpaa.2013.12.547 +[Abstract](44) +[PDF](1001.7KB)
Abstract:
We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in the two-dimensional Navier-Stokes equation, given by ${\mathcal L}_\alpha=-\Delta+|x|^2+\alpha \sigma(|x|)\partial_\theta$, where $\sigma(r)=r^{-2}(1-e^{-r^2})$, $\partial_\theta=x_1\partial_2-x_2\partial_1$ and $\alpha$ is a positive parameter tending to $+\infty$. We give a complete study of the resolvent of ${\mathcal L}_\alpha$ along the imaginary axis in the fast rotation limit $\alpha\to+\infty$ and we prove $\sup_{\lambda\in \mathbb{R}}\|({\mathcal L}_\alpha-i\lambda)^{-1}\|_{{\mathcal L}(\tilde L^2(\mathbb{R}^2))}\leq C\alpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.
On a nonlocal isoperimetric problem on the two-sphere
Ihsan Topaloglu
2013, 12(1): 597-620 doi: 10.3934/cpaa.2013.12.597 +[Abstract](33) +[PDF](1610.6KB)
Abstract:
In this article we analyze the minimization of a nonlocal isoperimetric problem (NLIP) posed on the $2$-sphere. After establishing the regularity of the free boundary of minimizers, we characterize two critical points of the functional describing (NLIP): the single cap and the double cap. We show that when the parameter controlling the influence of the nonlocality is small, the single cap is not only stable but also is the global minimizer of (NLIP) for all values of the mass constraint. In other words, in this parameter regime, the global minimizer of the (NLIP) coincides with the global minimizer of the local isoperimetric problem on the 2-sphere. Furthermore, we show that in certain parameter regimes the double cap is an unstable critical point.

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