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Long time dynamics for nonlinear wave-type equations with or without damping

Abstract : In this thesis, we study the qualitative behavior of solutions of nonlinear wave-type equations, with or without damping, putting a special emphasis on the description of the long-time dynamics of solutions in the energy space. Through the typical examples of the nonlinear Klein-Gordon equation (NLKG) with or without damping and the nonlinear wave equation (NLW), we study the behavior of solutions that decompose into a single soliton or into sums of solitons, when time goes to infinity.First, we show the conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity. Second, for the energy-critical NLW equation, we prove the existence of different types of multi-solitons based on the ground state or on suitable excited states, under various conditions on the space dimension and Lorentz speeds. Finally, we study the long-time dynamics of solitons and multi-solitons of the damped energy sub-critical NLKG equation.
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Submitted on : Friday, July 23, 2021 - 11:53:10 AM
Last modification on : Saturday, July 24, 2021 - 3:47:20 AM
Long-term archiving on: : Sunday, October 24, 2021 - 6:26:00 PM


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  • HAL Id : tel-03297223, version 1



Xu Yuan. Long time dynamics for nonlinear wave-type equations with or without damping. Analysis of PDEs [math.AP]. Institut Polytechnique de Paris, 2021. English. ⟨NNT : 2021IPPAX028⟩. ⟨tel-03297223⟩



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