Consider the initial value problem for the wave equation posed on the left half-line:

\begin{equation*}

\left\{\begin{aligned}

&u_{tt}- u_{xx}= 0 ,\qquad&&-\infty <x< 0\\

&u (x,0) = f(x), \qquad&&-\infty < x < 0 ,\\

&u_t(x,0)= g(x), \qquad&&-\infty < x < 0.

\end{aligned}\right.

\end{equation*}

Do the initial conditions uniquely determine the solution in the region $\{ (t,x): t \in \mathbb{R}, -\infty < x < 0 \}$? Explain your answer with convincing arguments.