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It is well known that a weak solution $\varphi$ to the initial boundary valueproblem 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 theparabolic boundary of $\Omega_T$, $\omega\in L^1(\Omega_T)$ with $\omega\geq0$, 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 upperbound 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*}

Logarithmic upper bounds for weak solutions to a class of parabolic equations