PROBLEM
The oscillations of an elastic string with damping are modelled by:
\begin{cases}
\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} + 2\gamma \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0 \text{ in } (0,L)\times(0,+\infty), \\
u(0,t) = u(L,t) = 0, t > 0, \\
u(x,0) = u_0(x), \frac{\partial u}{\partial t}(x,0) = u_1(x), 0 < x < L,
\end{cases}
where in (1): (a) $c$ and $\gamma$ are two positive constants. (b) $u_0$ and $u_1$ are two given
functions of $x$.
1) Show that if system (1) has a solution it is necessarily unique.
2) Assuming that $L = 1$, $c = 1$, $\gamma = 1$, and $u_0(x) = \sin \pi x$, $u_1(x) = 1$, $\forall x \in (0, 1)$, use
the method of separation of variables to solve the related system (1).
3) Same question than 2), excepted that the condition $\gamma = 1$ is replaced by $\gamma = \pi$.
