Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

Abstract

In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension. We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables. We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then, we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for noncharacteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step provides us with a result of independent interest: the trapping of the solution in self-similar variables near the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on a new analysis in the self-similar variable.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

“…See also the subcritical wave equation in dimension one where all blow-up profiles were found by Merle and Zaag [MZ07,MZ08] for general data (see the work of Caffarelli and Friedman for specific data [CF86]). …”

Blowup Solutions to the Semilinear Wave Equation with a Stylized Pyramid as a Blowup Surface

“…In this case x x 2 R. Using our techniques in [46] (in particular, the existence of a Lyapunov functional; see theorem 2 in that paper), we have…”

“…x .y; s/ D u. ; t / D e 22) translates into some estimate for w x , which will turn out to be in contradiction with the following bound we proved in[44,46]: 1 For all s log.T .x/ C 1/ C 1,…”

Dynamics of the focusing critical wave equation

“…Much more is known for pseudo-conformally subcritical wave equations (polynomial nonlinearity |u| p−1 u in space dimension N = 1, or, N ≥ 2 with p < N +3 N −1 ): see the works of Merle and Zaag e.g. [32,33]. In the energy-critical case there are numerical evidences that generic blow-up solutions behave like y 0 (t) see Bizoń, Chmaj and Tabor [3].…”