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Your search keyword '"Equations aux dérivées partielles"' showing total 2 results
Start Over Author "Equations aux dérivées partielles" Journal annales scientifiques de l'ecole normale superieure
2 results on '"Equations aux dérivées partielles"'

1. Long time dynamics for damped Klein-Gordon equations

Author

Geneviève Raugel, Wilhelm Schlag, Nicolas Burq, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics, University of Chicago, ANR: ANR-13-BS01-0010-03 (ANA\'E).NSF: DMS-1160817, and ANR-13-BS01-0010,ANAÉ,Analyse asymptotique des Equations aux dérivées partielles d'évolution(2013)

Subjects
Dynamical systems theory, General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Invariant manifold, Mathematics::Analysis of PDEs, Banach space, Dynamical Systems (math.DS), Type (model theory), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Dynamical Systems, 0101 mathematics, Klein–Gordon equation, Mathematical physics, Physics, 010102 general mathematics, Nonlinear system, Bounded function, symbols, MSC: 35L05, 74J20, 35H15, 37L30, 37L50, 010307 mathematical physics, Analysis of PDEs (math.AP), Sign (mathematics)
Abstract

For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifold theory in Banach spaces and convergence theorems).

Published
2017
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2. Elliptic estimates in composite media with smooth inclusions: an integral equation approach

Author

Eric Bonnetier, Faouzi Triki, Michael Vogelius, Habib Ammari, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), ANR-11-LABX-0025,PERSYVAL-lab,Systemes et Algorithmes Pervasifs au confluent des mondes physique et numérique(2011), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Elliptic curve, Integral representation, System of integral equations, Homogeneous, General Mathematics, Mathematical analysis, Composite number, Scalar (mathematics), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Composite media, Integral equation, Mathematics
Abstract

International audience; We consider a scalar elliptic equation for a composite medium consisting of homogeneous C^{1, α0} inclusions, 0< α0≤ 1, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functionsunctions converge in C^{0,α}, 0< α < α0 to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform C^{1,α} bounds on the solution, which are independent of the inter-inclusion distances.

Published
2015
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