Soliton resolution along a sequence of times with dispersive error for type II singular solutions to focusing energy critical wave equation

Abstract

In this paper, we study the soliton resolution conjecture for Type II singular solutions $\overrightarrow{u}(t)$ to the focusing energy critical wave equation in $R^d\times [0,T_{+})$, with $3\le d\le 5$. Suppose that $u$ has a singularity at $(x,t)=(0,T_{+})$, we show that along a sequence of times $t_n\uparrow T_{+}$ and in a neighborhood of $(0,T_{+})$, $\overrightarrow{u}(t_n)$ can be decomposed as the sum of a regular part in the energy space, a finite combination of modulated solitons (translation, scaling and Lorentz transform of steady states) and a residue term which goes to zero in the Strichartz norm. In addition, the residue term is asymptotically radial and can concentrate energy only in a thin annulus near $|x|=T_{+}-t_n$. Our main tools include a Morawetz estimate, similar to the one used for wave maps, and the use of a virial identity to eliminate dispersive energy in $|x|<\lambda (T_{+}-t_n)$ for any $\lambda \in (0,1)$.

1. A typo in Lemma 3.1 is corrected; 2. The dimension is updated to $3\le d\le 5$. The exclusion of $d=6$ is due to technical issues with the profile decomposition in this dimension. We thank Thomas Duyckaerts for pointing this out. See also arXiv:1601.01871, where the result is significantly strengthened. Only arXiv:1601.01871 will be submitted for publication

References in corpus (1)

• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces

Cited by in corpus (2)

• Universality of blow up profile for small blow up solutions to the energy critical wave map equation
• Quantization of time-like energy for wave maps into spheres