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

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 $(u_0^{\ast },u_1^{\ast })$, where ${\lim }_{t\to 0}\lambda (t)/t=0$ and $(u_0^{\ast },u_1^{\ast })\in {\dot{H}}^1\times L^2$. We construct a blow-up solution of this type such that $(u_0^{\ast },u_1^{\ast })$ is any pair of sufficiently regular functions with $u_0^{\ast }(0)>0$. 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.

39 pages; the new version takes into account the remarks of the referees

References in corpus (3)

• Type II blow up manifolds for the energy supercritical wave equation
• Near soliton dynamics and singularity formation for $L^2$ critical problems
• Non-dispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in R^3

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