A center-stable manifold for the energy-critical wave equation in ℝ³ in the symmetric setting

Abstract

Consider the focusing semilinear wave equation in R 3 with energy-critical nonlinearityThis equation admits stationary solutions of the form φ(x, a) := (3a) 1/4 (1 + a|x| 2 ) −1/2 , called solitons, which solve the elliptic equation −∆φ − φ 5 = 0.Restricting ourselves to the space of symmetric solutions ψ for which ψ(x) = ψ(−x), we find a local center-stable manifold, in a neighborhood of φ(x, 1), for this wave equation in the weighted Sobolev space x −1Ḣ 1 × x −1 L 2 . Solutions with initial data on the manifold exist globally in time for t ≥ 0, depend continuously on initial data, preserve energy, and can be written as the sum of a rescaled soliton and a dispersive radiation term. The proof is based on a new class of reverse Strichartz estimates, recently introduced by Beceanu and Goldberg and adapted here to the case of Hamiltonians with a resonance.

Global, Non-scattering solutions to the quintic, focusing semilinear wave equation on $\mathbb{R}^{1+3}$

“…The work [9] studied the dynamics of solutions with energy bounded above by a quantity slightly larger than the ground state energy. Stable manifolds for the critical wave equation in 3 spatial dimensions were constructed in [13] (studied further in [16]), [1], and [15]. Modulated soliton solutions emerging from randomized initial data were studied in [10].…”

Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space

“…In this paper we will consider the problem of stability for a non trivial stationary background. Our work is in the spirit of recent studies of asymptotic stability of solitary waves for semilinear wave equations (see for example [5,6,18,23,24]; see also [13,22,26,28] for finite time blow up regimes which correspond to asymptotic stability in suitable rescaled variables), but in a quasilinear setting. The background solution we choose is the catenoid, which is an embedded minimal surface in R 3 , and is a surface of revolution with topology S 1ˆR .…”

Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed

“…The Cauchy problem for the Zakharov-Kuznetsov equation is extensively studied in the literature. The global well-posedness of the Zakharov-Kuznetsov equation in H s (R × T L ) for s > 3 has been proved by Linares, Pastor and Saut [19] to study the transverse instability of the N-soliton of the Korteweg-de Vries equation. Molinet and Pilod [26] showed the global well-posedness in H 1 (R × T L ) by proving a bilinear estimate in the context of Bourgain's spaces X s,b .…”

“…Using the bi-linear estimate on Fourier restriction spaces in [26], the author controlled a loss of derivative for the nonlinearity of (1). For more results of center stable manifolds around relative equilibria, we refer to the papers [3,11,15,16,28], and the references therein.…”

“…Since the dominant order of the positive term of the Lyapunov function S c * (u) around the line solitary wave is 4, by using the dominant order of the positive term of S c * (u) we cannot control the error term which is the L 2 -inner product of P + u and L c * P − u. To obtain a sharper estimate of the error term, we construct invariant manifolds which is a Lipschitz graph function on the stable invariant space of the linearized operator with the Hölder exponent 3 2 < α < 2 at Q c * (x − c * t) (see the definition of G + l1,l2,α,δ,κ in Section 3 for the detail). By the order of the invariant manifolds at Q c * (x − c * t), we have that the order of unstable modes P + u on the invariant manifolds is α.…”