We are given that $u$ is the solution to an IVP with the following conditions: $$ \begin{cases} u_{tt} = u_{xx} \\ u_x(0, t) = 0 \\ u(x, 0) = u_0 \\ u_{t}(x, 0) = u_1 \end{cases} $$ We are also given that $u_0$ and $u_1$ are nonzero on the interval $[1, 2]$. And the task is to find all pairs of $(x,~ t)$ s. t. $u(x, t) = 0$.
I am having huge problems with this task, and, to be honest, with the PDEs as a subject in general, so I would also be very grateful if someone sent me useful materials on such problems and ways to understand this subject.
As for this problem, I tried finding general solution by applying d'Alambert's formula. However, found nothing useful.
With the Laplace transform, assuming $u(0,t)=0$
Solving the PDE
$$ s^2u(x,s)=u_{xx}(x,s)+u_1+s u_0,\ \ u_x(0,s)=0, u(0,s)=0 $$
we got
$$ u(x,s) = \frac{e^{-s x} \left(1-e^{s x}\right)^2 (s u_0+u_1)}{2 s^2} $$
with anti-transform
$$ u(x,t) = -\frac{1}{2} \theta (t-x) ((t-x)u_1+u_0)-\frac{1}{2} \theta (t+x) ((t+x)u_1+u_0)+t u_1+u_0 $$
Here $\theta(\cdot)$ is the Heaviside unit step function.