Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.

Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where $u_0$ is bounded initial data. Here either $V=H^1_0(\Omega)$ or $v=H^1(\Omega)$ and we take zero Neumann data.

How may I prove an $L^\infty-L^1$ smoothing effect of this form:

$$\lVert u(t) \rVert_{L^\infty} \leq Ct^{-\gamma}\lVert u_0 \rVert_{L^1}^\alpha$$ for all $t > 0$?

I have seen a proof when we have the porous medium equation, but it is complicated (even when picking the linear case) so I wondered if there is an easier method. Maybe something using Sobolev inequalities.