Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\in \Omega_T ,$$ $$u|_{\partial \Omega}=0,$$ $$u(0,\cdot)=u_0.$$ Assume that the initial condition $u_0 \in H^2(\Omega)\cap H^1_0(\Omega) \cap L^\infty(\Omega)$, can we prove that the solution $u$ is such that $u_t \in L^\infty(\Omega_T)$ ?

I found some old results which use much of regularity on $a$ and $u_0$ based on maximum principles. Is there any other ways to prove such results under weaker assumptions. Thank you.