18,891 research outputs found
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces
In this paper we consider the following Cauchy problem for the semi-linear
wave equation with scale-invariant dissipation and mass and power
non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta
u+\dfrac{\mu_1}{1+t} u_t+\dfrac{\mu_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x),
\,\, u_t(0,x)=u_1(x), \end{cases}\tag{} \end{align} where are nonnegative constants and . On the one hand we will prove a
global (in time) existence result for \eqref{CP abstract} under suitable
assumptions on the coefficients of the damping and the mass
term and on the exponent , assuming the smallness of data in exponentially
weighted energy spaces. On the other hand a blow-up result for \eqref{CP
abstract} is proved for values of below a certain threshold, provided that
the data satisfy some integral sign conditions. Combining these results we find
the critical exponent for \eqref{CP abstract} in all space dimensions under
certain assumptions on and . Moreover, since the global
existence result is based on a contradiction argument, it will be shown firstly
a local (in time) existence result
Semilinear Hyperbolic Equations in Curved Spacetime
This is a survey of the author's recent work rather than a broad survey of
the literature. The survey is concerned with the global in time solutions of
the Cauchy problem for matter waves propagating in the curved spacetimes, which
can be, in particular, modeled by cosmological models. We examine the global in
time solutions of some class of semililear hyperbolic equations, such as the
Klein-Gordon equation, which includes the Higgs boson equation in the Minkowski
spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. The crucial
tool for the obtaining those results is a new approach suggested by the author
based on the integral transform with the kernel containing the hypergeometric
function.\\ {\bf Mathematics Subject Classification (2010):} Primary 35L71,
35L53; Secondary 81T20, 35C15.\\ {\bf Keywords:} \small {de Sitter spacetime;
Klein-Gordon equation; Global solutions; Huygens' principle; Higuchi bound}Comment: arXiv admin note: text overlap with arXiv:1206.023
Wave and Klein-Gordon equations on hyperbolic spaces
We consider the Klein--Gordon equation associated with the Laplace--Beltrami
operator on real hyperbolic spaces of dimension ; as
has a spectral gap, the wave equation is a particular case of our
study. After a careful kernel analysis, we obtain dispersive and Strichartz
estimates for a large family of admissible couples. As an application, we prove
global well--posedness results for the corresponding semilinear equation with
low regularity data.Comment: 50 pages, 30 figures. arXiv admin note: text overlap with
arXiv:1010.237
Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds
We prove global weighted Strichartz estimates for radial solutions of linear
Schr\"odinger equation on a class of rotationally symmetric noncompact
manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces.
This yields classical Strichartz estimates with a larger class of exponents
than in the Euclidian case and improvements for the scattering theory. The
manifolds, whose volume element grows polynomially or exponentially at
infinity, are characterized essentially by negativity conditions on the
curvature, which shows in particular that the rich algebraic structure of the
Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive
properties of the equation. The proofs are based on known dispersive results
for the equation with potential on the Euclidean space, and on a new one, valid
for C^1 potentials decaying like 1/r^2 at infinity
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