It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation
$\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $
in
$\Omega _T\equiv \Omega \times (0,T)$
satisfies the uniform estimate
$$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$
provided that
$q \gt 1+{N}/{2}$
, where
Ω is a bounded domain in
${\open R}^N$
with Lipschitz boundary,
T > 0,
$\partial _p\Omega _T$
is the parabolic boundary of
$\Omega _T$
,
$\omega \in L^1(\Omega _T)$
with
$\omega \ges 0$
, and
λ is the smallest eigenvalue of the coefficient matrix
A. This estimate is sharp in the sense that it generally fails if
$q=1+{N}/{2}$
. In this paper, we show that the linear growth of the upper bound in
$\Vert f \Vert_{q,\Omega _T}$
can be improved. To be precise, we establish
$$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$