 Exponential drainage transform for porous-medium diffusion

The porous medium equation $h_t=\lambda (h{h}_x{)}_x-\Gamma h$ with constant $\Gamma >0$ becomes $H_{\tau }=\lambda (H{H}_x{)}_x$ under the displayed transformation. Substitution gives $h_t+\Gamma h=e^{-\Gamma t}{\tau }^{\prime }{H}_{\tau }$ and $\lambda (h{h}_x{)}_x=\lambda e^{-2\Gamma t}(H{H}_x{)}_x$, so ${\tau }^{\prime }=e^{-\Gamma t}$. This transformation applies to every solution with appropriately transformed initial and boundary conditions, not just a similarity solution. A one-sided constant-volume porous gravity current therefore has remaining physical volume $V{e}^{-\Gamma t}$ and a limiting front $[9\lambda V/(\varphi \Gamma ){]}^{1/3}$. The finite transformed time $\tau \to 1/\Gamma$ means that arbitrary initial profiles need not reach the universal large-time source shape before drainage removes them.