Consider the singular porous medium equation $$u_t - \Delta (|u|^{m-1}u) = 0$$ $$u(0)=u_0$$ given $u_0$ bounded, where $m \in (0,1)$.
When posed on $\mathbb{R}^n$, it is well known that mass is conserved when $m \in (m_c, 1)$ and mass is lost when $m \in (0,m_c)$, where $m_c$ is a number that depends on the dimension $n$.
When we pose this equation on a $n-$dimensional compact closed manifold, does this still hold true? Do we still get a range for which mass is lost, i.e., $$\int_M u(t,x)dx \neq \int_M u_0?$$
I've tried to find papers but there is not much on manifolds.
The solution satisfies $u \in C([0,T);L^2(M))$ with $|u|^{m-1}u \in L^2(0,T;H^1(M))$ and $$-\int_M u_0\phi(0) + \int_0^T \int_M (-u\phi_t + \nabla |u|^{m-1}u \nabla \phi) = 0$$ for all test functions vanishing at $t=T$. So it is not clear how to proceed?
