 Spatially nonintegrable forcing in transport

Smooth forcing need not be integrable in phase space. For the hyperbolic characteristic flow for an inverted oscillator, the source $x^2+v^2$ accumulates as $q_t=\frac{1}{2}\sinh (2t)(x^2+v^2)-(\cosh (2t)-1)xv$. Its smallest quadratic-form eigenvalue is $(1-e^{-2t})/2>0$ for $t>0$. Starting with a nonnegative integrable Gaussian function, the solution is therefore unbounded and outside every finite Lp space at positive times. Locally time-integrable $L^p$ forcing prevents this failure by the Minkowski integral inequality.