# wave equation with vanishing trace at infinity

Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^n$$. Consider the boundary value problem \label{pf0} \begin{aligned} \begin{cases} \Box u+qu=0\,\quad &\text{on (0,\infty)\times \Omega}, \\ u=f\,\quad &\text{on \Sigma=(0,\infty)\times \partial \Omega,}\\ (u,\partial_t u)\to 0 \,\quad &\text{ on \Omega as t\to \infty} \end{cases} \end{aligned}

Suppose that $$q \in C^{\infty}([0,\infty)\times \overline{\Omega})$$ and that $$f \in C^{\infty}_c((0,\infty)\times \partial \Omega)$$. Does the above boundary value problem admit a unique smooth solution?

Note that if initial data was assumed to be zero, this problem would be trivial by finite speed of propagation. However, I wonder if a similar technique could work when data vanishes at infinity instead. Uniqueness is the key issue here as existence is easy.

If $$q$$ is not signed, then in general the solution need not be unique.

The question of uniqueness can be reduced to the case where $$f \equiv 0$$.

In this case, the constant $$0$$ function obviously solve the PDE. So you just need an example of a non-zero solution.

Let $$v$$ be a Dirichlet eigenfunction of the Laplacian on $$\Omega$$ with eigenvalue $$\lambda$$, then (I'm not sure what your sign convention for $$\Box$$ is) your question has a special case $$u = \eta(t) v(x)$$ where now $$\eta$$ solves

$$\ddot{\eta} + (\lambda + q)\eta = 0$$

Given any smooth, non-vanishing function $$\eta : [0,\infty) \to \mathbb{R}$$ you can solve for a corresponding $$q$$.

In particular, you can choose to let $$\eta = 1/(1+t)$$ and require $$q = -\lambda + \frac{2}{(1+t)^2}$$

If $$q$$ has the correct sign, uniqueness should follow from energy estimates.

If you know more information about $$q$$, then there is a possibility to establish uniqueness on a case by case basis; but uniqueness would fail in the generality you just asked about.