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Register a new userExterior energy bounds for the critical wave equation close to the ground state
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By definition, the exterior asymptotic energy of a solution to a wave equation on $mathbb{R}^{1+N}$ is the sum of the limits as $tto pminfty$ of the energy in the the exterior ${|x|>|t|}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equation in odd space dimension $N$ is bounded from below by the conserved energy of the solution. In this article, we study the analogous problem for the linear wave equation with a potential begin{equation} label{abstractLW} tag{*} partial_t^2u+L_Wu=0,quad L_W:=-Delta -frac{N+2}{N-2}W^{frac{4}{N-2}} end{equation} obtained by linearizing the energy critical wave equation at the ground-state solution $W$, still in odd space dimension. This equation admits nonzero solutions of the form $A+tB$, where $L_WA=L_WB=0$ with vanishing asymptotic exterior energy. We prove that the exterior energy of a solution of eqref{abstractLW} is bounded from below by the energy of the projection of the initial data on the orthogonal complement of the space of initial data corresponding to these solutions. This will be used in a subsequent paper to prove soliton resolution for the energy-critical wave equation with radial data in all odd space dimensions. We also prove analogous results for the linearization of the energy-critical wave equation around a Lorentz transform of $W$, and give applications to the dynamics of the nonlinear equation close to the ground state in space dimensions $3$ and $5$.
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Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
Consider a finite energy radial solution to the focusing energy critical semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits type-II behavior, by which we mean that the critical Sobolev norm of the evolution stays bounded on the maximal interval of existence. We prove that along a sequence of times tending to the maximal forward time of existence, the solution decomposes into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume that the critical norm of the evolution localized to the light cone (the forward light cone in the case of global solutions and the backwards cone in the case of finite time blow-up) is less than 2 times the critical norm of the ground state solution W, then the decomposition holds without a restriction to a subsequence.
We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u_0,u_1)=E(W,0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to our previous work on radial Schrodinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.
Consider the energy-critical focusing wave equation in space dimension $Ngeq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution whose energy in the exterior ${|x|>|t|}$ of the wave cone vanishes asymptotically as $tto +infty$ and $tto -infty$. In a previous work (Cambridge Journal of Mathematics 2013, arXiv:1204.0031), we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, $0$ and $pm W$. This was crucial in the proof of soliton resolution in 3 space dimension. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behaviour as $rto infty$. We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag and the second author (Advances in Mathematics, 2015, arXiv:1409.3643) . We also study the propagation of the support of nonzero radial solutions with compactly supported initial data, and prove that these solutions cannot be nonradiative.
For the 5D energy-critical wave equation, we construct excited $N$-solitons with collinear speeds, i.e. solutions $u$ of the equation such that begin{equation*} lim_{tto+infty}bigg| abla_{t,x}u(t)- abla_{t,x}bigg(sum_{n=1}^{N}Q_{n}(t)bigg)bigg|_{L^{2}}=0, end{equation*} where for $n=1,ldots,N$, $Q_n(t,x)$ is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel-Merle and C^ote-Martel developed for the energy-critical wave and nonlinear Klein-Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.
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