Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

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

“…The solution of (1.1) corresponding to this profile is globally defined and nonscattering backward in time, satisfies a global bound similar to (1.7) for negative times, and is partially located around the light cone |t| = |x| as t → −∞. This type of solution is excluded by Proposition 7.1, using the nonexistence, shown in [KM08], of self-similar blow-up solutions of (1.1) which are compact up to scaling.…”

“…In particular, there cannot be small profiles, and the bound (1.7) implies that W is the only profile for this particular sequence t n . This yields the strong condition that the energy of the singular part E(a, ∂ t a) tends to E(W, 0) as t → 1 − (see Corollary 8.3), which can be combined with the results of [KM08] to complete the proof (see §8.3 and §8.4).…”

“…As is shown in Section 3, u may be decomposed as the sum of a solution to (1.1) which is well-defined around the blow-up time, and a singular part a(t, x) which is supported in the light cone |x| ≤ 1 − t. Choose a sequence t n → 1 − and a Bahouri-Gérard [BG99] profile decomposition associated to the sequence {(a(t n ), ∂ t a(t n )) n }. According to the result of [KM08], the bound (1.7) implies that there is one large profile and that the other profiles are small (see Remark 3.10). This contrasts with [KM08] where the minimality of the solution imposes automatically that there is only one profile (the large one).…”

“…According to the result of [KM08], the bound (1.7) implies that there is one large profile and that the other profiles are small (see Remark 3.10). This contrasts with [KM08] where the minimality of the solution imposes automatically that there is only one profile (the large one). In our case, we must show by another mechanism that the small profiles do not exist, which would imply by Theorem 2 that the large profile is W , yielding Theorem 1.…”

Blow-up Solutions of Quasilinear Hyperbolic Equations With Critical Sobolev Exponent

“…In our proof, we follow an idea from [8] where the problem (1.1) was studied when p = 2. To pave the way for the proof of Theorem 2.1, we shall need a number of lemmas.…”