Find solution $u=u(x,t)$ and describe domain, where it is uniquely defined

\begin{align}

&u_{tt}-u_{xx}=0,

\label{A}\\[2pt]

&u|_{t=x^2/2}= x^3,

\label{B}\\[2pt]

&u_t|_{t=x^2/2}= {\color{blue}{2}}x.

\label{C}

\end{align}

**Correction:** I replace $x$ by $2x$ in (\ref{C})