Sharp non-uniqueness of weak solutions to the semilinear wave equation

Abstract

In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation \[ \partial_{tt}u-\partial_{xx} u= Θ_1(\partial_t u)^2 + Θ_2(\partial_x u)^2. \] For any ${\Theta }_1{\Theta }_2<0$, $\beta <1$, we construct infinitely many weak solutions $u\in C_{t,x}^{\beta }$ with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity $C_{t,x}^{\beta }(\beta <1)$, the quadratic term is defined by Littlewood--Paley decomposition in $C_t{H}_x^{-s}$. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in $C_{t,x}^1$, this non-uniqueness result is sharp.