Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact (closed) smooth Riemannian manifold $M$, i.e., it satisfies $$\int_M u_0\varphi(0) =-\int_0^T\int_M u(t)\varphi'(t) +\int_0^T\int_M \nabla u^m(t) \cdot \nabla \varphi(t) $$ for all test functions $\varphi \in C^1([0,T]\times M)$ that vanish at $t=T$.
Suppose $u_0 \in L^\infty(M)$. It is the claimed in this work on top of page 10 that the $L^p$ norm decreases in time: $$\lVert u(t) \rVert_{L^p(M)} \leq \lVert u_0 \rVert_{L^p(M)}$$ for all $t$ and all $p$.
Formally this can be seen by: $$\frac{d}{dt}\int_M |u(t)|^p=p\int_M|u(t)|^{p-2}u(t)u_t(t)=p\int_M \nabla(|u(t)|^{p-2}u(t))\cdot \nabla(u^m(t))\\ =...=-pm(p-1)\int_M |u(t)|^{p+m-3}|\nabla u(t)|^2 \leq 0.$$ Note that the weak formulation is used in the second equality.
Obviously this calculation is formal because $u$ is not that smooth. How do I make it rigorous? Edit: let us assume $u(t) \in L^\infty(M)$ if necessary.
