Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5

TL;DR

This paper constructs specific type II blow-up solutions for the energy-critical wave equation in five dimensions, demonstrating precise blow-up rates and the influence of asymptotic profiles on solution behavior.

Contribution

It introduces a method to construct type II blow-up solutions with prescribed asymptotic profiles and specific concentration rates in the energy-critical wave equation in dimension five.

Findings

Constructed solutions with concentration rate t(t)  t^4

Provided examples with t(t)  t^{ u + 1} for t > 8

Showed the influence of asymptotic profiles on blow-up dynamics

Abstract

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of the family of solitons $W_\lambda$, asymptotically decomposes in the energy space as a sum of a bubble $W_\lambda$ and an asymptotic profile $(u_0^*, u_1^*)$, where $\lim_{t\to 0}\lambda(t)/t = 0$ and $(u^*_0, u^*_1) \in \dot H^1\times L^2$. We construct a blow-up solution of this type such that $(u^*_0, u^*_1)$ is any pair of sufficiently regular functions with $u_0^*(0) > 0$. For these solutions the concentration rate is $\lambda(t) \sim t^4$. We also provide examples of solutions with concentration rate $\lambda(t) \sim t^{\nu + 1}$ for $\nu > 8$, related to the behaviour of the asymptotic profile near the origin.