Threshold for blowup for the supercritical cubic wave equation

Abstract

In this paper, we discuss singularity formation for the focusing cubic wave equation in the energy supercritical regime. For this equation an explicit nontrivial self-similar blowup solution was recently found by the first and third author in [27]. In the seven dimensional case it was proven to be stable along a co-dimension one manifold of initial data. Here, we provide numerical evidence that this solution is in fact a critical solution at the threshold between finite-time blowup and dispersion. Furthermore, we discuss the spectral problem arising in the stability analysis in general dimensions d ≥ 5.

On blowup for the supercritical quadratic wave equation

“…In the description of threshold dynamics for energy supercritical wave equations, self-similar solutions appear to play the key role. This has been observed numerically for power-type nonlinearities [7,20], but also for more physically relevant models such as wave maps [6,2] or the Yang-Mills equation in equivariant symmetry [3,5]. We note that the latter reduces essentially to a radial quadratic wave equation in d ≥ 7, hence Eq.…”

“…However, very recently, the first explicit candidate for a self-similar threshold solution has been found by the second and third author in [21] for the focusing cubic wave equation in all supercritical space dimensions d ≥ 5. In d = 7, by the conformal symmetry of the linearized equation, the genuine unstable direction could be given in closed form, see also [20], which allowed for a rigorous stability analysis. Interestingly, the same effect occurs for the quadratic wave equation and the new self-similar solution (1.4) in d = 9, which explains the specific choice of the space dimension in Theorem 1.1.…”

“…(1.1), the relevant similarity profiles appear to be the trivial one (1.2) and its first non-trivial "excitation". Namely, numerical work on supercritical power nonlinearity wave equations in the radial case [7,20] yields evidence that generic blowup is described by the ODE profile, while the threshold separating generic blowup from global existence is given by the stable manifold of the first excited profile, see also [4]. The first step in showing such genericity results would be to establish stability of the ODE profile, and show that its first excitation is co-dimension one stable (which indicates that the stable manifold splits the phase space locally into two connected components).…”

Dynamics at the threshold for blowup for supercritical wave equations outside a ball

“…In addition, there exists a countable family of unstable selfsimilar solutions which correspond to non-generic finite time blowups [4] and the unique self-similar solution with exactly one unstable direction was shown numerically to be critical in the sense that its codimension-one stable manifold separates dispersive and singular solutions [2]. Recently, the analogous critical dynamics at the threshold for blowup has been analysed for the cubic wave equation in higher dimensions [5]; remarkably, for d 5 the critical selfsimilar solution has been found explicitly which allowed the authors to prove rigorously its codimension-one stability [6].…”

On blowup for the supercritical quadratic wave equation

“…We note that the rescaling (1-3) leaves invariant the energy norm P H 1 ‫ޒ.‬ d / L 2 ‫ޒ.‬ d / of .u.t; /; @ t u.t; // precisely when d D 6, in which case (1-1) is referred to as energy critical. In this case, it can be easily shown that in addition to (1-2) no other radial and smooth self-similar solutions to (1-1) exist; see [Kavian and Weissler 1990]. However, in the energy supercritical case, i.e., for d 7, numerics [Kycia 2011] indicate that in addition to (1-2) there are nontrivial, radial, globally defined, smooth, and decaying similarity profiles.…”