Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for the porous medium equation, in which also $u^m \in L^2(0,T;H^1(\Omega))$.)

I am seeking a estimate of the form $$\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$$ for $h >0$ small (want to send $h$ to zero later). Here $C$ is some constant not depending on $h$, $t$ or $u$. Note that the power of $h$ must be one, not a half.

This type of estimate is useful in order to show time regularity.

**My question is, what techniques are available and can one use to in order to derive such an estimate, apart from the one I describe below?**

I know of one way, which is detailed in the book on PME by Vazquez. This way comes out of rescaling the solution so that it satisfies the PME again with a rescaled initial data and then using the comparison principle after picking the scaling parameter appropriately.

Is there some other method? I'd prefer variational methods and not semigroup.

Why I want to do this? Because I work on a similar such equation where the rescaling I described above doesn't work so I am looking for different methods.