Let $u(x,y)$ be e a solution of the equation $$u_{tt}-u_{xx}=q(t,x)u$$ with initial conditions that are all zero. If $q$ is bounded, then $u\equiv 0$.
My attempt: the solution of the homogeneous is zero, so, using Duhamel, the solution is $$u(t,x)= \frac{1}{2}\int \int_{\Delta(t,x)} q(r,s)u(r,s)\text{d}r \text{d}s$$
where $\Delta(t,x)$ is the triangle limited by the characteristic lines containing the point $(t,x)$. If $M = \sup |q|$, then $$|u(t,x)|\leq \frac{M}{2}\int \int_{\Delta(t,x)} |u(r,s)|\text{d}r \text{d}s. $$ If $S(t,x) = \sup_{\Delta(t,x)}|u|$, then $$|u(t,x)|\leq \frac{M \cdot S(t,x)}{2}\int \int_{\Delta(t,x)} \text{d}r \text{d}s = \frac{M \cdot S(t,x) \cdot t^2}{2}.$$ Is it correct? If it is, how can I go from here?