= Solution
Choose $f_0(x,v)=e^{-(x^2+v^2)}$ and $h(t,x,v)=x^2+v^2$. The initial <Gaussian function> is smooth and belongs to every finite <Lp space>, and is also bounded. The source is smooth and nonzero. The accumulated source along the backward <characteristic curve> is
$$
\begin{aligned}
q_t(x,v)&=\int_0^t|A_{s-t}(x,v)|^2\,ds\\
&=\frac{\sinh(2t)}2(x^2+v^2)-(\cosh(2t)-1)xv.
\end{aligned}
$$
Its quadratic-form eigenvalues are $(e^{2t}-1)/2$ and $(1-e^{-2t})/2$. Both are positive for $t>0$, so
$$
f_t(x,v)=e^{-|A_{-t}(x,v)|^2}+q_t(x,v)
\geq\frac{1-e^{-2t}}2(x^2+v^2).
$$
Consequently \b[$\|f_t\|_p=\infty$ for every $t>0$ and every finite $p\geq1$]; it is unbounded, so its $L^\infty$ <norm> is infinite as well. This <spatially nonintegrable forcing in transport> example avoids any ambiguity about whether the last endpoint is included in “all $p$”.
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