Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation
251
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider the nonlinear damped Klein-Gordon equation [ partial_{tt}u+2alphapartial_{t}u-Delta u+u-|u|^{p-1}u=0 quad text{on} [0,infty)times mathbb{R}^N ] with $alpha>0$, $2 le Nle 5$ and energy subcritical exponents $p>2$. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times. We first prove that the nonlinear object is necessarily a bound state. Next, we show that when the nonlinear object is a non-degenerate state or a degenerate excited state satisfying a simplicity condition, the convergence holds for all positive times, with an exponential or algebraic rate respectively. Last, we provide an example where the solution converges exactly at the rate $t^{-1}$ to the excited state.
rate research
Read More
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.
Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. In a previous paper, we proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a solution with the compactness property, that is a solution whose trajectory stays in a compact subset of the energy space up to space translation and scaling. It is conjectured that the only solutions with the compactness property are stationary solutions and solitary waves that are Lorentz transforms of the former. In this note we prove this conjecture with an additional non-degeneracy assumption related to the invariances of the elliptic equation satisfied by stationary solutions. The proof uses a standard monotonicity formula, modulation theory, and a new channel of energy argument which is independent of the space dimension.
This article resolves some errors in the paper Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions.
The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations of second order and their solutions by a singularity analysis are classified. In particular, it has been shown that whenever they have the Painleve property, they can be transformed to standard forms by Moebius transformations of dependent variable and arbitrary smooth transformations of independent variable whose solutions, depending on the values of parameters, are expressible in terms of either elementary functions or Jacobi elliptic functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
