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Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 3

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Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$
J. Colliander, Justin Holmer, Monica Visan and Xiaoyi Zhang
2008, 7(3): 467-489 doi: 10.3934/cpaa.2008.7.467 +[Abstract](462) +[PDF](272.2KB)
Abstract:
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori $L_{t,x}^8$ spacetime control on solutions.
Internal nonnegative stabilization for some parabolic equations
B. E. Ainseba and Sebastian Aniţa
2008, 7(3): 491-512 doi: 10.3934/cpaa.2008.7.491 +[Abstract](453) +[PDF](243.0KB)
Abstract:
The internal zero-stabilization of the nonnegative solutions to some parabolic equations is investigated. We provide a necessary and a sufficient condition for nonnegative stabilizability in terms of the sign of the principal eigenvalue of a certain elliptic operator. This principal eigenvalue is related to the rate of the convergence of the solution. We give some evaluations of this principal eigenvalue with respect to the geometry of the domain and of the support of the control. A stabilization result for an age-dependent population dynamics with diffusion is also established.
On the decay in time of solutions of some generalized regularized long waves equations
Youcef Mammeri
2008, 7(3): 513-532 doi: 10.3934/cpaa.2008.7.513 +[Abstract](646) +[PDF](238.0KB)
Abstract:
We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation $u_t + u_x +u^\rho u_x + H(u_{x t})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{x x t} +\partial_x^{-1}u_{y y} =0$. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power $\rho$ of the nonlinearity is higher than $3$, the respective solution of these equations tends to zero when time rises with a decay rate of order close to $\frac{1}{2}$.
Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators
Roberta Bosi, Jean Dolbeault and Maria J. Esteban
2008, 7(3): 533-562 doi: 10.3934/cpaa.2008.7.533 +[Abstract](554) +[PDF](369.2KB)
Abstract:
By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the $L^2$ norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.
Spectral analysis and limit behaviours in a spring-mass system
M. Pellicer and J. Solà-Morales
2008, 7(3): 563-577 doi: 10.3934/cpaa.2008.7.563 +[Abstract](512) +[PDF](219.8KB)
Abstract:
We consider a model for a damped spring-mass system that is a strongly damped wave equation with dynamic boundary conditions. In a previous paper we showed that for some values of the parameters of the model, the large time behaviour of the solutions is the same as for a classical spring-mass damper ODE. Here we use spectral analysis to show that for other values of the parameters, still of physical relevance and related to the effect of the spring inner viscosity, the limit behaviours are very different from that classical ODE.
Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system
Ling Hsiao, Fucai Li and Shu Wang
2008, 7(3): 579-589 doi: 10.3934/cpaa.2008.7.579 +[Abstract](723) +[PDF](172.8KB)
Abstract:
The combined quasineutral and inviscid limit for the Vlasov-Poisson-Fokker-Planck (VPFP) system is rigorously derived in this paper. It is shown that the solution of VPFP system converges to the solution of incompressible Euler equations with damping. The proof of convergence result is based on compactness arguments and the so-called relative-entropy method.
Regularity for solutions of the two-phase Stefan problem
Marianne Korten and Charles N. Moore
2008, 7(3): 591-600 doi: 10.3934/cpaa.2008.7.591 +[Abstract](470) +[PDF](152.1KB)
Abstract:
We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.
On periodic elliptic equations with gradient dependence
Massimiliano Berti, M. Matzeu and Enrico Valdinoci
2008, 7(3): 601-615 doi: 10.3934/cpaa.2008.7.601 +[Abstract](545) +[PDF](212.6KB)
Abstract:
We construct entire solutions of $\Delta u=f(x,u,\nabla u)$ which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.
A fourth order nonlinear degenerate parabolic equation
Changchun Liu
2008, 7(3): 617-630 doi: 10.3934/cpaa.2008.7.617 +[Abstract](572) +[PDF](174.6KB)
Abstract:
In this paper, we study a fourth order degenerate parabolic equation, which arises in epitaxial growth of nanoscale thin films. We establish the existence of weak solutions, based on the uniform estimates for the approximate solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.
Regularity for positive weak solutions to semi-linear elliptic equations
Li Ma and Lin Zhao
2008, 7(3): 631-643 doi: 10.3934/cpaa.2008.7.631 +[Abstract](583) +[PDF](183.8KB)
Abstract:
In this paper, we first derive a new non-linear type inequality for Newtonian potential and then we study the regularity problem for positive weak solutions to the non-linear Laplace equation:

$-\Delta u= f(u)\quad$ in $\Omega,$

with $f(u)\in L^1(\Omega)$. Here $\Omega$ is a bounded domain in $R^n$, and $f(u)$ is a regular function with respect to $u$. We give an apriori estimate for positive weak solutions. We show that under some appropriate assumptions on the non-linear term $f$, the positive weak solutions are in fact in some local Sobolev space $W_{l o c}^{1,\tau}(\Omega)$. We also derive a very general local monotonicity formula for variational solutions to the equation above with special nonlinear term $f$.

A fixed point result with applications in the study of viscoplastic frictionless contact problems
Mircea Sofonea, Cezar Avramescu and Andaluzia Matei
2008, 7(3): 645-658 doi: 10.3934/cpaa.2008.7.645 +[Abstract](580) +[PDF](219.3KB)
Abstract:
Let $C(\mathbb R_+;X)$ denote the Fréchet space of continuous functions defined on $\mathbb R_+=[0,\infty)$ with values on a real Banach space $(X,$ ||$\cdot$||$_X)$. We prove a fixed point theorem for operators $\Lambda:C(\mathbb R_+;X)\to C(\mathbb R_+;X)$ which satisfy a sequence of inequalities involving an integral term. Then we consider a mathematical model which describes the frictionless contact between a viscoplastic body and a deformable foundation. The process is quasistatic and is studied on the unbounded interval of time $[0,\infty)$. We provide the variational formulation of the problem, then we use the abstract fixed point theorem to prove the existence of a unique weak solution to the model. We complete our study with a regularity result.
Mathematical analysis of a PDE epidemiological model applied to scrapie transmission
Najat Ziyadi, Said Boulite, M. Lhassan Hbid and Suzanne Touzeau
2008, 7(3): 659-675 doi: 10.3934/cpaa.2008.7.659 +[Abstract](1345) +[PDF](216.8KB)
Abstract:
The aim of this paper is to analyse a dynamic model which describes the spread of scrapie in a sheep flock. Scrapie is a transmissible spongiform encephalopathy, endemic in a few European regions and subject to strict control measures. The model takes into account various factors and processes, including seasonal breeding, horizontal and vertical transmission, genetic susceptibility of sheep to the disease, and a long and variable incubation period. Therefore the model, derived from a classical SI (susceptible-infected) model, also incorporates a discrete genetic structure for the flock, as well as a continuous infection load structure which represents the disease incubation. The resulting model consists of a set of partial differential equations which describe the evolution of the flock with respect to time and infection load. To analyse this model, we use the semigroup and evolution family theory, which provides a flexible mathematical framework to determine the existence and uniqueness of a solution to the problem. We show that the corresponding linear model has a unique classical solution and that the complete nonlinear model has a global solution.
Analysis of a biosensor model
Walter Allegretto, Yanping Lin and Zhiyong Zhang
2008, 7(3): 677-698 doi: 10.3934/cpaa.2008.7.677 +[Abstract](451) +[PDF](287.3KB)
Abstract:
In this paper we consider a biosensor model in $R^3$, consisting of a coupled parabolic differential equation with Robin boundary condition and an ordinary differential equation. Theoretical analysis is done to show the existence and uniqueness of a Holder continuous solution based on a maximum principle, weak solution arguments. The long-time convergence to a steady state is also discussed as well as the system situation. Next, a finite volume method is applied to the model to obtain an approximate solution. Drawing in part on the analytical results given earlier, we establish the existence, stability and error estimates for the approximate solution, and derive $L^2$ spatial norm convergence properties. Finally, some illustrative numerical simulation results are presented.
On the periodic Schrödinger-Debye equation
Alexander Arbieto and Carlos Matheus
2008, 7(3): 699-713 doi: 10.3934/cpaa.2008.7.699 +[Abstract](541) +[PDF](207.4KB)
Abstract:
We study local and global well-posedness of the initial value problem for the Schrödinger-Debye equation in the periodic case. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new a priori estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this a priori estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$ without any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.
Multiple solutions for critical elliptic systems in potential form
Emmanuel Hebey and Jérôme Vétois
2008, 7(3): 715-741 doi: 10.3934/cpaa.2008.7.715 +[Abstract](555) +[PDF](393.9KB)
Abstract:
We discuss and prove existence of multiple solutions for critical elliptic systems in potential form on compact Riemannian manifolds.
Errata
Daniele Bartolucci and Luigi Orsina
2008, 7(3): 743-744 doi: 10.3934/cpaa.2008.7.743 +[Abstract](354) +[PDF](83.7KB)
Abstract:
We correct the statements of Proposition 4.1 and Theorem 4.1 in [1]: "Uniformly elliptic Liouville type equations: Concentration compactness and a priori estimates," Comm. Pure and Appl. Analysis, 4 (2005), 499--522.

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