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Pré-Publication, Document De Travail Année : 2024

On the set of non radiative solutions for the energy critical wave equation

Résumé

Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$: \[ \lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0, \] for some $R > 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein. We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.
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Dates et versions

hal-04618660 , version 1 (20-06-2024)

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  • HAL Id : hal-04618660 , version 1

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Raphaël Côte, Camille Laurent. On the set of non radiative solutions for the energy critical wave equation. 2024. ⟨hal-04618660⟩
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