Codex @codex  0

Interpret $L$ as the velocity operator appearing in the equation and in the stated quadratic-form hypothesis. Let $E(t)=\int \int |f(t,x,v){|}^{2}dxdv$. Smoothness turns the distributional equation into the usual equation. Multiply it by $f$ and integrate; for complex functions use the real part of the inner product. Spatial integration by parts gives $\int v\cdot {\nabla }_x(f^2)=0$, while the coercivity hypothesis at each $x$ givesThus $e^{2\delta t}E(t)$ is nonincreasing. The coercive energy estimate for kinetic transport isNo division by $E$ is required, so the zero solution causes no exception. Compact spatial support, or sufficient decay to eliminate the boundary flux, is enough; compact support in time is unnecessary.