  It is well known that a weak solution $\varphi$ to the initial boundary value
problem for the uniformly parabolic equation
$\partial_t\varphi-\mbox{div}(A\nabla \varphi) +\omega\varphi= f $ in
$\Omega_T\equiv\Omega\times(0,T)$ satisfies the uniform estimate
  $$
  \|\varphi\|_{\infty,\Omega_T}\leq
\|\varphi\|_{\infty,\partial_p\Omega_T}+c\|f\|_{q,\Omega_T}, \ \ \
  c=c(N,\lambda, q, \Omega_T),
  $$ provided that $q>1+\frac{N}{2}$, where $\Omega$ is a bounded domain in
$\mathbb{R}^N$ with Lipschitz boundary, $T>0$, $\partial_p\Omega_T$ is the
parabolic boundary of $\Omega_T$, $\omega\in L^1(\Omega_T)$ with $\omega\geq
0$, and $\lambda$ is the smallest eigenvalue of the coefficient matrix $A$.
This estimate is sharp in the sense that it generally fails if
$q=1+\frac{N}{2}$. In this paper we show that the linear growth of this upper
bound in $\|f\|_{q,\Omega_T}$ can be improved. To be precise, we establish
  \begin{equation*}
  \|\varphi\|_{\infty,\Omega_T}\leq
  \|\varphi_0\|_{\infty,\partial_p\Omega_T}+c\|f\|_{1+\frac{N}{2},\Omega_T}\left(\ln(\|f\|_{q,\Omega_T}+1)+1\right).
  \end{equation*}
