[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-965870","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n  \u003Cdiv class=\u0022row\u0022\u003E\n    \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n      \u003Ch2 class=\u0022h1 text-left\u0022\u003E  Equation de Klein-Gordon nonlin\u00e9aire amortie\n\u003C\/h2\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n          3\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemester:\n          Semestre 2\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ELecture:\u003C\/span\u003E\n                                      \u003Cspan\u003E18\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                                                                        \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EFran\u00e7ais\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003Enon\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n    \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EPr\u00e9requis\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EConnaissances de math\u00e9matiques g\u00e9n\u00e9rales. \u00c9quations diff\u00e9rentielles lin\u00e9aires et non lin\u00e9aires. Bases de l\u2019analyse fonctionnelle. \u00c9quations aux d\u00e9riv\u00e9es partielles lin\u00e9aires.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cul\u003E\n\u003Cli\u003ELe probl\u00e8me de Cauchy local en temps pour l\u2019\u00e9quation de Klein-Gordon non lin\u00e9aire avec amortissement (T. Cazenave et A. Haraux).\u003C\/li\u003E\n\u003Cli\u003EToute solution globale est born\u00e9e par la m\u00e9thode d\u2019\u00e9nergie (T. Cazenave).\u003C\/li\u003E\n\u003Cli\u003ELes solitons de l\u2019\u00e9quation de Klein-Gordon non lin\u00e9aire (incluant l\u2019\u00e9tude spectrale de l\u2019op\u00e9rateur lin\u00e9aris\u00e9).\u003C\/li\u003E\n\u003Cli\u003ELe principe de concentration-compacit\u00e9 (P.-L. Lions)\u003C\/li\u003E\n\u003Cli\u003EPremier r\u00e9sultat de convergence ; \u00e9mergence de solitons pour des sous-suites de temps (E. Feireisl).\u003C\/li\u003E\n\u003Cli\u003E\u201cModulation\u201d des solitons et \u00e9quations des param\u00e8tres g\u00e9om\u00e9triques (position et vitesse de chaque soliton)\u003C\/li\u003E\n\u003Cli\u003E\u00c9tude d\u2019un syst\u00e8me de dimension fini de type Toda (F. Merle et H. Zaag).\u003C\/li\u003E\n\u003Cli\u003ERaffinement du r\u00e9sultat de compacit\u00e9, convergence pour toute suite de temps, propri\u00e9t\u00e9 des signes altern\u00e9s, positions asymptotiques des solitons (R. C\u00f4te, Y. Martel et X. Yuan)\u003C\/li\u003E\n\u003C\/ul\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EObjectifs d\u0026#039;apprentissage\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ECe cours est consacr\u00e9 \u00e0 l\u2019\u00e9tude de l\u2019\u00e9mergence de solitons (ondes progressives) dans le comportement global de toutes les solutions globales de l\u2019\u00e9quation de Klein-Gordon non lin\u00e9aire avec un amortissement, en dimension 1 d\u2019espace.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n                                \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EBibliographie\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ET. Cazenave, Uniform estimates for solutions of nonlinear Klein-Gordon equations, Journal of Functional Analysis, 60 (1985), 36-55.\nT. Cazenave and A. Haraux, An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications, 13. The Clarendon Press, Oxford University Press, New York, 1998.\nR. C\u00f4te, Y. Martel, X. Yuan, Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation, Arch. Ration. Mech. Anal. 239, 1837\u20131874 (2021). https:\/\/arxiv.org\/abs\/2002.01826\nE. Feireisl, Finite energy travelling waves for nonlinear damped wave equations, Quart. Appl. Math., 56 (1998), 55\u201370.\nP.-L. Lions, On positive solutions of semilinear elliptic equations in unbounded domains, Nonlinear diffusion equations and their equilibrium states, II (Berkeley, CA, 1986), 85\u2013122, Math. Sci. Res. Inst. Publ., 13, Springer, New York, 1988.\nF. Merle and H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012) n. 3, 581\u2013648.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EEvaluation Continue Int\u00e9grale\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n  \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]