On the -decay and local energy decay of solutions to nonlinear Klein-Gordon equations
Philip Brenner (2003)
Banach Center Publications
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Philip Brenner (2003)
Banach Center Publications
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Serge Alinhac (2005)
Bulletin de la Société Mathématique de France
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We investigate for which metric (close to the standard metric ) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (, non integrable) decay conditions on ; in particular, decays like along wave cones.
Thierry Gallay, Romain Joly (2009)
Annales scientifiques de l'École Normale Supérieure
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We consider the damped wave equation on the whole real line, where is a bistable potential. This equation has travelling front solutions of the form which describe a moving interface between two different steady states of the system, one of which being the global minimum of . We show that, if the initial data are sufficiently close to the profile of a front for large , the solution of the damped wave equation converges uniformly on to a travelling front as . The proof of this...
Anton Savostianov, Sergey Zelik (2014)
Mathematica Bohemica
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We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of with damping terms of the form , where or . The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when . For existence of smooth attractors is more complicated...
Nalini Anantharaman, Matthieu Léautaud (2012)
Journées Équations aux dérivées partielles
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This article is a proceedings version of the ongoing work [
Cécile Huneau (2014-2015)
Séminaire Laurent Schwartz — EDP et applications
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In this note, we discuss the nonlinear stability in exponential time of Minkowski space-time with a translation space-like Killing field, proved in [13]. In the presence of such a symmetry, the vacuum Einstein equations reduce to the Einstein equations with a scalar field. We work in generalized wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in [13] is due to the decay in of free solutions...
Grzegorz Karch (2000)
Studia Mathematica
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Large time behavior of solutions to the generalized damped wave equation for is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where , , and the nonlinear term is either or . In this case, the asymptotic profile of solutions...
Sergey A. Nazarov, Jari Taskinen (2011)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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We study the linearized water-wave problem in a bounded domain (a finite pond of water) of , having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point of the water surface, where a submerged body touches the surface (see Fig. 1)....
Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)
Bollettino dell'Unione Matematica Italiana
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This paper deals with the well-posedness in of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.
Jean-Luc Joly, Guy Métivier, Jeffrey Rauch (1998-1999)
Séminaire Équations aux dérivées partielles
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The nonlinear dissipative wave equation in dimension has strong solutions with the following structure. In the solutions have a focusing wave of singularity on the incoming light cone . In that is after the focusing time, they are smoother than they were in . The examples are radial and piecewise smooth in
Jérôme Adou, Adama Coulibaly, Narcisse Dakouri (2013)
Annales mathématiques Blaise Pascal
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A pneumatic tyre in rotating motion with a constant angular velocity is assimilated to a torus whose generating circle has a radius . The contact of the tyre with the ground is schematized as an ellipse with semi-major axis . When and (where is the velocity of the sound), we show that at the rapid time scale , the air motion within a torus periodically excited on its surface generates an acoustic wave . A study of this acoustic wave is conducted and shows that the mode associated...
Marino Badiale, Vieri Benci, Sergio Rolando (2007)
Journal of the European Mathematical Society
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We prove the existence of cylindrical solutions to the semilinear elliptic problem , , , where , and has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.
Michael W. Smiley (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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The semilinear differential equation (1), (2), (3), in with , (a nonlinear wave equation) is studied. In particular for , the existence is shown of a weak solution , periodic with period , non-constant with respect to , and radially symmetric in the spatial variables, that is of the form . The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative...
Gabriel Rivière (2014)
Annales de l’institut Fourier
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We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of -damped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic...
Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)
Journal of the European Mathematical Society
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We study the Cauchy problem in the hyperbolic space for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space new phenomena arise, which depend on the properties of the diffusion process in . We also investigate a family of travelling wave solutions,...