I am trying to understand the behaviour of solutions of parabolic equations as $t\to 0^+$. More specifically: $\mathbb T^d$ being the $d$-dimensional flat torus, let $u_0\in L^\infty(\mathbb T^d)$ and consider the solution $u$ of the heat equation \begin{cases} \partial_t u-\Delta u=0& \\ u(0,\cdot)=u_0.\end{cases} We know that there exists a unique weak solution $u\in L^2(0,T;H^1(\mathbb T^d))$ that further satisfies: for any $p\in [1;\infty)$, $u\in C([0,T];L^p(\mathbb T^d))$. In particular $$ \Vert u(t,\cdot)-u_0\Vert_{L^p}\underset{t\to 0^+}\rightarrow 0.$$ Of course if $u_0$ is not continuous, one can not take $p=\infty$.
My question is the following: is there an "Egorov type theorem", in the sense that, for any $\epsilon>0$, there exists a measurable subset $E_\epsilon$ such that $|\mathbb T^d\backslash E_\epsilon|\leq \epsilon$, and such that $u(t,\cdot)$ converges as $t\to 0^+$ uniformly to $u_0$ in $E_\epsilon$?
So far I have only tried but without success the following:
1-Trying to extend the proof of the Egorov theorem to this uncountable case, using the continuity in $L^p$, but without success.
2-Trying to approximate $u_0$ by a sequence $(u_{0,k})$ of smooth initial data, and using the Egorov theorem on this sequence, but I have also failed there.