Bounded solution for parabolic equation 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. Are there any other ways to prove such results under weaker assumptions?
 A: For the heat equation $Lu=u_t-\Delta u=0$ to guarantee boundedness of $u_t$ as $t$ tends to zero one has to demand more regularity from the initial function, e.g. $u_0\in C^{1,1}(\bar\Omega)$ (the first order derivatives are uniformly Lipschitz in $\Omega$).  
As for the low order term, differentiating wrt $t$ we have that $u_t$ satisfies the same equation. So if $u_0,\Delta u_0\in L_\infty(\Omega)$ then $u_t|_{t=0}\in L_\infty(\Omega)$ and $u_t$ is bounded.
Denote $a_0=\|a\|_{L_\infty(\Omega)}$ and $v(t)=\|u\|_{L_\infty(\Omega_t)}$. For the first BVP $u_t-\Delta u=f$, $u|_{t=0}=u_0$ with zero boundary condition it follows that 
$$
v(t)\le t a_0 v(t)+\|u_0\|_{L_\infty(\Omega)}.
$$
From here for $T_0=1/(2a_0)$ it follows that $v(T_0)\le 2\|u_0\|_{L_\infty(\Omega)}$. For arbitrary $T$ the estimate $v(T)\le C\|u_0\|_{L_\infty(\Omega)}$ follows from step by step argument, where $C$ depends upon $T$ as well as on $a_0$. For the derivative it gives the estimate 
$$
\|u_t\|_{L_\infty(\Omega_T)}\le C(T,a_0)(\|u_0\|_{L_\infty(\Omega)}+\|\Delta u_0\|_{L_\infty(\Omega)}).
$$
