wave equation with vanishing trace at infinity

Question

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem \begin{equation}\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} \end{equation}

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.

Answer 1

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 $$ q = -\lambda + \frac{2}{(1+t)^2}$$

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

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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.