Consider a homogeneous, linear, first-order PDE
$$L u \equiv \left( \sum_{i = 1}^d f^i(x) \frac{\partial}{\partial x^i} + c(x) \right) u(x) = 0$$
on some compact domain $\Omega \subset \mathbb{R}^d$. Obviously this system always has $u = 0$ as a solution; my question is what sorts of conditions on the coefficients $f^i(x)$ and $c(x)$ are sufficient in order to guarantee that the zero solution is unique subject to the boundary condition $u|_{\partial \Omega} = 0$.
I know that well-posedness of first-order PDEs is usually studied via the method of characteristics, but as I understand that's typically useful in thinking of the PDE as an initial value problem in which boundary conditions are specified on an initial-value surface and evolved from there. Because here I'm treating the system as a Dirichlet problem, the inhomogenous problem $Lu = g$, $u|_{\partial \Omega} = h$ may not in general be well-posed; but that's OK because I just care about uniqueness of the zero solution to the homogeneous problem.
I have one partial result from Oleinik and Radkevic (https://www.springer.com/gp/book/9781468489675), which consider second-order linear PDEs with nonnegative characteristic form, of which the equation I gave above is a special case (since its characteristic form is identically zero). Then from e.g. Theorem 1.6.2 of this book I can conclude that the zero solution is unique if $c^* < 0$ in $\Omega \cup \partial \Omega$, where $c^* \equiv c – \sum_{i = 1}^d \partial_i f^i$ is the zero-derivative term of the adjoint $L^*$ of $L$. But because the operator $L$ I care about is genuinely a first-order operator, while the condition $c^* < 0$ comes from considering second-order operators, I imagine there must be much more general sufficient conditions for the uniqueness of the zero solution than just $c^* < 0$.
Best Answer
The method of characteristics looks like the right way to solve this. Along paths that satisfy ${\rm d}x_i/{\rm d}t = f_i(\vec{x})$, one finds $u(\vec{x}(t))$ evolves according to ${\rm d}u/{\rm d}t = -c u$. If the path terminates at $\partial\Omega$, then $u(x) = 0$ along the whole path. This leads to our first necessary condition for the existence of a nonzero solution:
(1) $\exists$ path $\vec{x}(t)$ satisfying ${\rm d}x_i/{\rm d}t = f_i(\vec{x})$ with origin and terminus (limits as $t \rightarrow \pm\infty$) in the interior of $\Omega$.
For a continuous $u(\vec{x})$, the value of $u(\vec{x}(t))$ cannot diverge when $t \rightarrow \pm\infty$. Excepting a set of measure zero, all paths $\vec{x}(t)$ start at a repulsor and end at an attractor (rather than, say, a saddle point). Two more necessary conditions for the existence of a nonzero solution are therefore:
(2) $c < 0$ at $\vec{x}(-\infty)$
(3) $c > 0$ at $\vec{x}(+\infty)$
Except for a set of measure zero, we can probably assume these inequalities are strict, i.e. $c < 0$ and $c > 0$, respectively (convergence is possible for $c = 0$ but not guaranteed, depending on derivative terms). With the strict inequalities, conditions (1-3) are also sufficient for nonzero solutions $u(\vec{x})$ to exist. That can be seen as follows:
Starting with a point $\vec{x}_0$ along the path $\vec{x}(t)$, define a size-$\epsilon$ cross section (orthogonal to the streamlines of ${\rm d}x_i/{\rm d}t = f_i(\vec{x})$) and posit that $u(\vec{x})$ varies smoothly from $u(x_0) = 1$ to $u = 0$ at the boundaries of the cross section. The value of $u(\vec{x})$ along the "past" and "future" of this cross section is obtained by propagating along the characteristics using ${\rm d}u/{\rm d}t = -c u$. All these characteristics originate from the same repulsor (where $u = 0$) and terminate at the same attractor (also where $u = 0$). Fill in the rest of $\Omega$ with the null solution $u = 0$. Thus we have constructed a nonzero, continuous-valued solution to the PDE.
There are a bunch of singular edge cases where the necessary and sufficient conditions don't coincide, i.e. if $\lVert f \rVert = u = 0$ at the same point (fixable by rescaling $f$ and $u$), if $\lVert f\rVert = 0$ over an open subset of $\Omega$, if $\lVert f\rVert = 0$ on the boundary $\partial\Omega$, if $c = 0$ at $\vec{x}(\pm\infty)$. In the space of possible functions $(\vec{f}, u)$, these singular cases only occur in a set of measure zero, so are not very interesting. Almost everywhere, conditions (1-3) are both necessary and sufficient.
Putting this another way, we can say (almost everywhere) that the zero solution is unique if:
Coming back to your condition $c^* < 0$: Note that $\partial_i f^i < 0$ at attractors (this always holds, regardless of whether it's a node, limit cycle, toroid, chaotic attractor, etc.). Therefore, if $c^* < 0$ on $\Omega$, it follows that $c = c^* + \partial_i f^i < 0$ at all of the attractors. Therefore, the second condition above is always satisfied when $c^* < 0$. The condition above is the more general sufficient (and necessary) condition for uniqueness (with the caveats noted above).
Since any dynamical system can be represented by ${\rm d}x_i/{\rm d}t = f_i(\vec{x})$ and dynamical systems can be really, really complicated, the general condition can be hard to work with, so more specific conditions like $c^* < 0$ might be more useful.
Also, defining the value of $c$ is tricky when the attractor / repulsor isn't a point. Taking the average over limit cycles is straightforward, chaotic attractors less so (ergodic theory).