Vol. 7, No. 2, 2014

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Nondispersive decay for the cubic wave equation

Roland Donninger and Anıl Zenginoğlu

Vol. 7 (2014), No. 2, 461–495
Abstract

We consider the hyperboloidal initial value problem for the cubic focusing wave equation

$\left(-{\partial }_{t}^{2}+{\Delta }_{x}\right)v\left(t,x\right)+v{\left(t,x\right)}^{3}=0,\phantom{\rule{1em}{0ex}}x\in {ℝ}^{3}.$

Without symmetry assumptions, we prove the existence of a codimension-4 Lipschitz manifold of initial data that lead to global solutions in forward time which do not scatter to free waves. More precisely, for any $\delta \in \left(0,1\right)$, we construct solutions with the asymptotic behavior

$\parallel v-{v}_{0}{\parallel }_{{L}^{4}\left(t,2t\right){L}^{4}\left({B}_{\left(1-\delta \right)t}\right)}\lesssim {t}^{-\frac{1}{2}+}$

as $t\to \infty$, where ${v}_{0}\left(t,x\right)=\sqrt{2}∕t$ and ${B}_{\left(1-\delta \right)t}:=\left\{x\in {ℝ}^{3}:|x|<\left(1-\delta \right)t\right\}$.

Keywords
nonlinear wave equations, soliton resolution conjecture, hyperboloidal initial value problem, Kelvin coordinates
Mathematical Subject Classification 2010
Primary: 35L05, 58J45, 35L71
Secondary: 35Q75, 83C30