Optimal bounds on positive blow-up solutions for a semilinear wave equation

By C. Antonini, F. Merle. 2001

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Determination of the blow-up rate for a critical semilinear wave equation (2005, background): “1: The proof is the same as in the subcritical case [3], except for one integration by parts where the boundary term can no longer be neglected.”
On growth rate near the blowup surface for semilinear wave equations (2005, methodology): “Our method relies on the one hand, on the estimates in similarity variables introduced in Antonini and Merle [2] and used in [16, 17], and, on the other hand, on covering techniques adapted to the geometric shape of the blowup surface, which are the new ingredients of this paper.”
Determination of the blow-up rate for the semilinear wave equation (2003, methodology): “Let us remark that Theorem 1′ follows from Theorem 1 and the change of variables (3) as in [2].”
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation (2006, methodology): “Fix now t0 ∈ [1, 2] and recall, from the beginning of the proof that ∂te(zd,R)(x, t) = ∑N j=1 ∂xj (∂xjzd,R∂tzd,R).”
Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension (2007, background): “Proof of Proposition 2.1: See Antonini and Merle [4] and Appendix A.”
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation (2008, methodology): “We first recall some dispersion estimates from [3] and [16]: Lemma 3.”