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

In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation $${\mathrm{\partial }}_{tt}u-{\mathrm{\partial }}_{xx}u={\mathrm{\Theta }}_1({\mathrm{\partial }}_{t}u{)}^2+{\mathrm{\Theta }}_2({\mathrm{\partial }}_{x}u{)}^2\mathrm{.}$$ For any ${\mathrm{\Theta }}_1{\mathrm{\Theta }}_2<0$, $\beta <1$, 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.