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Communications on Pure & Applied Analysis

2005 , Volume 4 , Issue 2

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Global solutions of the wave-Schrödinger system with rough data
Takafumi Akahori
2005, 4(2): 209-240 doi: 10.3934/cpaa.2005.4.209 +[Abstract](31) +[PDF](428.2KB)
Abstract:
We prove that the wave-Schröodinger system is globally well-posed for data in $(H^{s_{1}} \times \dot{H}^{s_{2}} \times \dot{H}^{s_{2}-1})(\mathbb^{d})$, where $d=3,4$ and $s_{1},s_{2}> q_{d} \ (q_{3}=(\sqrt{57}-5)/4, \ q_{4}=\sqrt{3}-1)$. Our proof is based on the I-method. We introduce the space $\Omega^{s,b}$ which controls the low frequency part and the modified multiplier for I-method to work in the space $\Omega^{s,b}$.
Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients
Gui-Qiang Chen and  Kenneth Hvistendahl Karlsen
2005, 4(2): 241-266 doi: 10.3934/cpaa.2005.4.241 +[Abstract](49) +[PDF](333.4KB)
Abstract:
We study the well-posedness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit $(t,x)$--dependence:

$\partial_tu + \sum_{i=1}^d\partial_{x_i}f_i(u,t,x)=\sum_{i,j=1}^d\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u),$

where $a(u,t,x)=(a_{ij}(u,t,x))=\sigma^a(u,t,x)\sigma^a(u,t,x)^\top$ is nonnegative definite and each $x\mapsto f_i(u,t,x)$ is Lipschitz continuous. We establish a well-posedness theory for the Cauchy problem for such degenerate parabolic equations via Kruzkov's device of doubling variables, provided $\sigma^a(u,t,\cdot)\in W^{2,\infty}$ for the general case and the weaker condition $\sigma^a(u,t,\cdot)\in W^{1,\infty}$ for the case that $a$ is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions.

Approximations of degree zero in the Poisson problem
C. Davini and  F. Jourdan
2005, 4(2): 267-281 doi: 10.3934/cpaa.2005.4.267 +[Abstract](27) +[PDF](374.5KB)
Abstract:
We discuss a technique for the approximation of the Poisson problem under mixed boundary conditions in spaces of piece-wise constant functions. The method adopts ideas from the theory of $\Gamma$-convergence as a guideline. Some applications are considered and numerical evaluation of the convergence rate is discussed.
On positive solutions of the elliptic sine-Gordon equation
Goong Chen , Zhonghai Ding and  Shujie Li
2005, 4(2): 283-294 doi: 10.3934/cpaa.2005.4.283 +[Abstract](27) +[PDF](1152.9KB)
Abstract:
The aim of this paper is to study positive solutions of the elliptic sine-Gordon equation on a bounded domain with homogeneous Dirichlet boundary condition, which models the steady state of the Josephson $\pi-$junction in superconductivity. The properties of positive solutions are investigated theoretically and numerically.
On homogenization of nonlinear hyperbolic equations
Y. Efendiev and  B. Popov
2005, 4(2): 295-309 doi: 10.3934/cpaa.2005.4.295 +[Abstract](28) +[PDF](243.9KB)
Abstract:
In this paper we study homogenization of nonlinear hyperbolic equations. The weak limit of the solutions is investigated by approximating the flux functions with piecewise linear functions. We study mostly Riemann problems for layered velocity fields as well as for the heterogeneous divergence free velocity fields.
Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature
Y. Goto , K. Ishii and  T. Ogawa
2005, 4(2): 311-339 doi: 10.3934/cpaa.2005.4.311 +[Abstract](33) +[PDF](353.5KB)
Abstract:
A new proof of the convergence of the Bence-Merriman-Osher algorithm for the motion of mean curvature is given. The idea is making use of the approximate distance function to the interface and analogous argument in the singular limiting problem for the Allen-Cahn equation via an auxiliary function given by the primitive function of the heat kernel.
A result on singularly perturbed elliptic problems
Andrés Ávila and  Louis Jeanjean
2005, 4(2): 341-356 doi: 10.3934/cpaa.2005.4.341 +[Abstract](32) +[PDF](302.2KB)
Abstract:
We consider a class of equations of the form

$ -\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1(\mathbf R^N).$

For a local minimum $x_0$ of the potential $V(x)$, we show that there exists a sequence $\varepsilon_n\to 0$, for which corresponding solutions $u_n(x) \in H^1(\mathbf R^N) $ concentrate at $x_0$. Our assumptions on $f(\xi)$ are mainly the ones under which the associated autonomous problem

$ -\Delta v + V(x_0)v = f(v), \quad v\in H^1(\mathbf R^N),$

admits a non trivial solution.

On the pointwise jump condition at the free boundary in the 1-phase Stefan problem
Donatella Danielli and  Marianne Korten
2005, 4(2): 357-366 doi: 10.3934/cpaa.2005.4.357 +[Abstract](32) +[PDF](223.4KB)
Abstract:
In this paper we obtain the jump (or Rankine-Hugoniot) condition at the interphase for solutions in the sense of distributions to the one phase Stefan problem $u_t= \Delta (u-1)_+.$ We do this by approximating the free boundary with level sets, and using methods from the theory of bounded variation functions. We show that the spatial component of the normal derivative of the solution has a trace at the free boundary that is picked up in a natural sense. The jump condition is then obtained from the equality of the $n$-density of two different disintegrations of the free boundary measure. This is done under an additional condition on the $n$-density of this measure. In the last section we show that this condition is optimal, in the sense that its satisfaction depends on the geometry of the initial data.
Existence and stability of periodic travelling-wavesolutions of the Benjamin equation
Jaime Angulo Pava and  Borys Alvarez Samaniego
2005, 4(2): 367-388 doi: 10.3934/cpaa.2005.4.367 +[Abstract](30) +[PDF](348.5KB)
Abstract:
A family of steady periodic water waves in very deep fluids when the surface tension is present and satisfying the following nonlinear pseudo-differential equation $ u_t + u u_x + u_{x x x} +l \mathcal{H} u_{x x}=0$, known as the Benjamin equation, is shown to exist. Here $\mathcal{H}$ denotes the periodic Hilbert transform and $l \in\mathbb{R}$. By fixing a minimal period we obtain, via the implicit function theorem, an analytic curve of periodic travelling-wave solutions depending on the parameter $l$. Moreover, by making some changes in the abstract stability theory developed by Grillakis, Shatah, and Strauss, we prove that these travelling waves are nonlinearly stable to perturbations with the same wavelength.
Uniformly distributed points on the sphere
Jorge Rebaza
2005, 4(2): 389-403 doi: 10.3934/cpaa.2005.4.389 +[Abstract](19) +[PDF](415.8KB)
Abstract:
In this work, we present uniformly distributed sequences on the unit sphere, and we show that this property is equivalent to requiring the sequences to have a low discrepancy. Numerical integration over the sphere is taken as a direct application, and the corresponding errors are estimated. Special care is taken in relating these concepts and properties to those for the euclidean case. Several examples of uniformly distributed sequences of nodes (ensembles) are presented.
On two classes of generalized viscous Cahn-Hilliard equations
Riccarda Rossi
2005, 4(2): 405-430 doi: 10.3934/cpaa.2005.4.405 +[Abstract](43) +[PDF](393.6KB)
Abstract:
This paper investigates two classes of generalized viscous Cahn-Hilliard equations , featuring two different laws for the mobility, which is assumed to depend on the chemical potential. Both equations can be obtained with the new derivation of equations of Cahn-Hilliard type proposed by M.E. GURTIN [14]. Approximation and compactness tools allow to prove well-posedness and, in one case, regularity results for the equations supplemented with initial and suitable boundary conditions.
Global solvability for a singular nonlinear Maxwell's equations
W. Wei and  H. M. Yin
2005, 4(2): 431-444 doi: 10.3934/cpaa.2005.4.431 +[Abstract](26) +[PDF](217.2KB)
Abstract:
In this paper we study a singular nonlinear evolution system:

$\frac{\partial}{\partial t}[\mu(x,|\mathbf H|)\mathbf H]+ \nabla\times [r(x,t) \nabla \times \mathbf H]=\mathbf F(x,t),$

where $\mathbf H$ represents the magnetic field in a quasi-stationary electromagnetic field and $\mu(x,|\mathbf H|)$ is the magnetic permeability in a conductive medium, which strongly depends on the strength of $\mathbf H$ such as $\mu(x,|\mathbf H|)=|\mathbf H|^b$ with $b>0$. We prove that under appropriate initial and boundary conditions the system has a global weak solution and the solution is also unique.

Hopf bifurcations of ODE systems along the singular direction in the parameter plane
Ruyuan Zhang
2005, 4(2): 445-461 doi: 10.3934/cpaa.2005.4.445 +[Abstract](22) +[PDF](212.0KB)
Abstract:
This paper considers Hopf bifurcation of ordinary differential systems along the singular direction in the parameter plane. Hopf Bifurcation of the same system along non-singular directions have been studied recently in [9]. As an application of our main results, we also obtain results of existence and non-existence for a type of degenerate Hopf bifurcation (i.e., the type without transversality) for one parameter ODE systems.
Asymptotic behavior of solutions to the nonlinear breakage equations
Lie Zheng
2005, 4(2): 463-473 doi: 10.3934/cpaa.2005.4.463 +[Abstract](32) +[PDF](164.9KB)
Abstract:
The nonlinear breakage equations are a mathematical model for the dynamics of cluster growth when clusters undergo binary collisions resulting either in coalescence or breakup with possible transfer of matter. Each of these two events may happen with an a priori prescribed probability depending for instance on the sizes of the colliding clusters. The model consists of a countable number of non-locally coupled nonlinear ordinary differential equations modeling the concentration of the various clusters. In the present paper we consider asymptotic behavior of solutions as time tends to infinity and prove the weak* convergence to steady states provided at least two monoclusters appear after a collision, and the weak* convergence to some equilibrium state at the expense of making stronger assumptions. A result on the strong convergence has also been obtained.
Forced periodic solutions for piezoelectric crystals
Giovanni Cimatti
2005, 4(2): 475-485 doi: 10.3934/cpaa.2005.4.475 +[Abstract](22) +[PDF](221.3KB)
Abstract:
The system of partial differential equations governing the dynamics and, in the stationary case, the electro-elastic equilibrium of a piezoelectric crystal when a given electric potential is applied on the surface, is studied in connection with the aspects of existence, uniqueness and regularity of solutions. We prove that the corresponding semigroup is in fact a group of isometries. When the data are periodic functions we also provide a condition for the existence of forced periodic vibrations in both the damped and undamped case.

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