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

Abstract

Abstract. Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions.Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.

Global dynamics away from the ground state for the energy-critical nonlinear wave equation

“…The first two conditions together with K(W ) = 0 imply that 4) and so ϕ n is bounded inḢ 1 ⊂ L 2 * ∩ rL 2 by the Sobolev and Hardy inequalities. We deal with possible concentration by the dyadic decomposition in x ∈ R d :…”

“…This problem arises after applying the one-pass theorem. Fortunately, we will see that the blowup analysis by Duyckaerts, Merle, Kenig [3,4] precludes it, so that we can proceed essentially in the same way as in the subcritical case.…”

“…Note that the solutions constructed in [12] in the three-dimensional case belong to this tube. Moreover, Duyckaerts, Kenig, Merle [3] showed that all type-II blowup (i.e., blowup with bounded energy norm) under the constraint (1.9) is of the form of those solutions found in [12]. But the tube around S might also contain solutions which do not blow up but rather scatter to S. This would correspond to the center-stable manifolds in [15,16].…”

“…We remark that the dimensional restriction is needed only for using the blow-up characterization by Duyckaerts-Kenig-Merle [4].…”

“…The key step is to prove the "one-pass" theorem, which states that the transition from the scattering region to the blow-up region can take place at most once along each trajectory. The main new ingredients are the control of the scaling parameter and the blow-up characterization by Duyckaerts, Kenig and Merle [3,4]. …”