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

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*}