The question is as follows:
The function $u(\mathbf x,t)$ defined on an open connected domain $\Omega \subset \Re ^2$ satisfies $$u_{tt} - \Delta u + qu = \phi, \ \ \mathbf x\in\Omega; t\gt 0 \\ \cfrac{\partial u}{\partial n} + \beta u = 0, \ \ \mathbf x \in\partial\Omega; t\gt 0 \\ u(\mathbf x, 0) = f; \ \ u_t(\mathbf x,0)=0, $$ where $f(\mathbf x)$ and $q(\mathbf x)$ are smooth and uniformly bounded in $\Omega$, and $\beta \gt 0$ is constant. Show that if such a problem has a solution $u\in C^0(\bar{\Omega})$, the solution must be unique.
I am also given the following hint: Note that the procedure to prove uniqueness fails because $q$ is allowed to become negative somewhere in $\Omega$. Consider instead the problem for $v(\mathbf x, t)$, which is related to $u(\mathbf x, t)$ via $u=e^{rt}v$, choosing $r$ appropriately to make the energy integral work.
So far I have worked out $u_{tt}$ and $\Delta u$ to form the following PDE: $$ (r^2 + q)v +2rv_t + v_{tt} - \Delta v =\phi e^{-rt} $$
I am not really sure how to go on from here, I know I must choose a value for $r$ but I don't know how. Any help would be much appreciated, thanks in advance!
