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Communications on Pure & Applied Analysis

2015 , Volume 14 , Issue 4

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T. Ogawa and  Tohru Ozawa
2015, 14(4): i-iii doi: 10.3934/cpaa.2015.14.4i +[Abstract](27) +[PDF](162.5KB)
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Professor Gustavo Ponce on the occasion of his sixtieth birthday.

Gustavo Ponce was born on April 20, 1952, in Venezuela. He received his B.A. in 1976 from Universidad Central de Venezuela and his Ph. D. in 1982 with the dissertation entitled ``Long time stability of solutions of nonlinear evolution equations" under the supervision of Sergiu Klainerman and Louis Nirenberg at Courant Institute, New York University. After professional experiences at University of California at Berkely (1982-1984), Universidad Central de Venezuela (1984-1986), University of Chicago (1986-1989), and Pennsylvania State University (1989-1991), he was appointed to a full professorship at Department of Mathematics, University of California at Santa Barbara in 1991, where he has remained up until now.
A remark on the well-posedness of a degenerated Zakharov system
Vanessa Barros and  Felipe Linares
2015, 14(4): 1259-1274 doi: 10.3934/cpaa.2015.14.1259 +[Abstract](24) +[PDF](398.5KB)
We extend the local well-posedness theory for the Cauchy problem associated to a degenerated Zakharov system. The new main ingredients are the derivation of Strichartz and maximal function norm estimates for the linear solution of a Schrödinger type equation with missing dispersion in one direction. The result here improves the one in [10].
Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations
Thomas Duyckaerts , Carlos E. Kenig and  Frank Merle
2015, 14(4): 1275-1326 doi: 10.3934/cpaa.2015.14.1275 +[Abstract](28) +[PDF](773.3KB)
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces
G. Fonseca , G. Rodríguez-Blanco and  W. Sandoval
2015, 14(4): 1327-1341 doi: 10.3934/cpaa.2015.14.1327 +[Abstract](25) +[PDF](386.5KB)
We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polynomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.
On a system of semirelativistic equations in the energy space
Kazumasa Fujiwara , Shuji Machihara and  Tohru Ozawa
2015, 14(4): 1343-1355 doi: 10.3934/cpaa.2015.14.1343 +[Abstract](41) +[PDF](372.2KB)
Well-posedness of the Cauchy problem for a system of semirelativistic equations is shown in the energy space. Solutions are constructed as a limit of an approximate solutions. A Yudovitch type argument plays an important role for the convergence arguments.
Modified wave operators without loss of regularity for some long range Hartree equations. II
Jean Ginibre and  Giorgio Velo
2015, 14(4): 1357-1376 doi: 10.3934/cpaa.2015.14.1357 +[Abstract](42) +[PDF](447.3KB)
We continue the study of the theory of scattering for some long range Hartree equations with potential $|x|^{-\gamma}$, performed in a previous paper, denoted as I, in the range $1/2 < \gamma < 1$. Here we extend the results to the range $1/3 < \gamma < 1/2$. More precisely we study the local Cauchy problem with infinite initial time, which is the main step in the construction of the modified wave operators. We solve that problem without loss of regularity between the asymptotic state and the solution, as in I, but in contrast to I, we are no longer able to cover the entire subcriticality range of regularity of the solutions. The method is an extension of that of I, using a better approximate asymptotic form of the solutions obtained as the next step of a natural procedure of successive approximations.
Nonlinear dispersive wave equations in two space dimensions
Nakao Hayashi , Seishirou Kobayashi and  Pavel I. Naumkin
2015, 14(4): 1377-1393 doi: 10.3934/cpaa.2015.14.1377 +[Abstract](29) +[PDF](411.2KB)
We study the global existence and time decay of solutions to nonlinear dispersive wave equations $ \partial_t^2 u+\frac{1}{\rho^2}( -\Delta) ^{\rho }u=F ( \partial _t u )$ in two space dimensions, where $F( \partial _t u) =\lambda \vert \partial _t u\vert ^{p-1}\partial _t u$ or $\lambda \vert \partial _t u \vert ^p$, $\lambda \in \mathbf{C,}$ with $ p > 2 $ for $0 < \rho <1,$ $p > 3$ for $\rho =1,$ and $p > 1+\rho $ for $1 < \rho <2.$ If $\rho =1,$ then the equation converts into the well-known nonlinear wave equation.
Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$
Tsukasa Iwabuchi and  Kota Uriya
2015, 14(4): 1395-1405 doi: 10.3934/cpaa.2015.14.1395 +[Abstract](32) +[PDF](386.0KB)
We are concerned with the ill-posedness issue for the nonlinear Schrödinger equation with the quadratic nonlinearity $|u|^2$ and prove the norm inflation in the dimensions $1 \le n \le 3$. This is the extension of the ill-posed result by Kishimoto-Tsugawa [12] in one dimension and also the remaining case of Iwabuchi-Ogawa [7].
Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms
Boyan Jonov and  Thomas C. Sideris
2015, 14(4): 1407-1442 doi: 10.3934/cpaa.2015.14.1407 +[Abstract](24) +[PDF](712.6KB)
The existence of global small $\mathcal O(\varepsilon )$ solutions to quadratically nonlinear wave equations in three space dimensions under the null condition is shown to be stable under the simultaneous addition of small $\mathcal O(\nu)$ viscous dissipation and $\mathcal O(\delta)$ non-null quadratic nonlinearities, provided that $\varepsilon \delta/\nu\ll 1$. When this condition is not met, small solutions exist ``almost globally'', and in certain parameter ranges, the addition of dissipation enhances the lifespan.
A numerical approach to Blow-up issues for Davey-Stewartson II systems
Christian Klein and  Jean-Claude Saut
2015, 14(4): 1443-1467 doi: 10.3934/cpaa.2015.14.1443 +[Abstract](45) +[PDF](4257.0KB)
We provide a numerical study of various issues pertaining to the dynamics of the Davey-Stewartson II systems. In particular we investigate whether or not the properties (blow-up, radiation,...) displayed by the focusing and defocusing Davey-Stewartson II integrable systems persist in the non integrable cases.
On the pointwise decay estimate for the wave equation with compactly supported forcing term
Hideo Kubo
2015, 14(4): 1469-1480 doi: 10.3934/cpaa.2015.14.1469 +[Abstract](30) +[PDF](358.9KB)
In this paper we derive a new type of pointwise decay estimates for solutions to the Cauchy problem for the wave equation in 2D, in the sense that one can diminish the weight in the time variable for the forcing term if it is compactly supported in the spatial variables. As an application of the estimate, we also establish an improved decay estimate for the solution to the exterior problem in 2D.
A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation
Satoshi Masaki
2015, 14(4): 1481-1531 doi: 10.3934/cpaa.2015.14.1481 +[Abstract](36) +[PDF](724.4KB)
This article is concerned with time global behavior of solutions to focusing mass-subcritical nonlinear Schrödinger equation of power type with data in a critical homogeneous weighted $L^2$ space. We give a sharp sufficient condition for scattering by proving existence of a threshold solution which does not scatter at least for one time direction and of which initial data attains minimum value of a norm of the weighted $L^2$ space in all initial value of non-scattering solution. Unlike in the mass-critical or -supercritical case, ground state is not a threshold. This is an extension of previous author's result to the case where the exponent of nonlinearity is below so-called Strauss number. A main new ingredient is a stability estimate in a Lorenz-modified-Bezov type spacetime norm.
Remarks on global solutions of dissipative wave equations with exponential nonlinear terms
Makoto Nakamura
2015, 14(4): 1533-1545 doi: 10.3934/cpaa.2015.14.1533 +[Abstract](24) +[PDF](420.1KB)
The Cauchy problem for dissipative wave equations with exponential type nonlinear terms is considered in the energy space in two spatial dimensions. The nonlinear terms have a singularity at the origin, and global solutions are shown based on the Gagliardo-Nirenberg type inequality.
Remarks on the full dispersion Davey-Stewartson systems
Caroline Obrecht and  J.-C. Saut
2015, 14(4): 1547-1561 doi: 10.3934/cpaa.2015.14.1547 +[Abstract](39) +[PDF](424.0KB)
We consider the Cauchy problem for the Full Dispersion Davey-Stewartson systems derived in [23] for the modeling of surface water waves in the modulation regime and we investigate some of their mathematical properties, emphasizing in particular the differences with the classical Davey-Stewartson systems.
Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data
Tadahiro Oh
2015, 14(4): 1563-1580 doi: 10.3934/cpaa.2015.14.1563 +[Abstract](27) +[PDF](512.1KB)
We consider the Cauchy problem for the defocusing nonlinear Schrödinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS with limit periodic functions as initial data under some regularity assumption.
The dynamics of vortex filaments with corners
Luis Vega
2015, 14(4): 1581-1601 doi: 10.3934/cpaa.2015.14.1581 +[Abstract](23) +[PDF](1145.5KB)
This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the so-called binormal flow. The case of a regular polygon studied in collaboration with F. de la Hoz is also considered.

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