Nonscattering solutions and blowup at infinity for the critical wave equation

Details Nonscattering solutions and blowup at infinity for the critical wave equation Nonscattering solutions and blowup at infinity for the critical wave equation

2012-01-16

arxiv.org/abs/1201.3258v1

Mathematics

Analysis of PDEs

51 pages

Scientific paper

We consider the critical focusing wave equation $(-\partial_t^2+\Delta)u+u^5=0$ in $\R^{1+3}$ and prove the existence of energy class solutions which are of the form \[ u(t,x)=t^\frac{\mu}{2}W(t^\mu x)+\eta(t,x) \] in the forward lightcone $\{(t,x)\in\R\times \R^3: |x|\leq t, t\gg 1\}$ where $W(x)=(1+\frac13 |x|^2)^{-\frac12}$ is the ground state soliton, $\mu$ is an arbitrary prescribed real number (positive or negative) with $|\mu|\ll 1$, and the error $\eta$ satisfies \[ \|\partial_t \eta(t,\cdot)\|_{L^2(B_t)} +\|\nabla \eta(t,\cdot)\|_{L^2(B_t)}\ll 1,\quad B_t:=\{x\in\R^3: |x|

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