Nondispersive decay for the cubic wave equation

Abstract

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

$$\left(-{\partial }_t^2+{\Delta }_x\right)v\left(t,x\right)+v{\left(t,x\right)}^3=0,x\in {ℝ}^3.$$

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 $B_{\left(1-\delta \right)t}:=\left\{x\in {ℝ}^3:\left|x\right|<\left(1-\delta \right)t\right\}$.