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

It is well known that a weak solution $\phi$ to the initial boundary value problem for the uniformly parabolic equation ${\mathrm{\partial }}_t\phi -div(A\mathrm{\nabla }\phi )+\omega \phi =f$ in ${\mathrm{\Omega }}_T\equiv \mathrm{\Omega }\times (0,T)$ satisfies the uniform estimate $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty },{\mathrm{\Omega }}_T}\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty },{\mathrm{\partial }}_p{\mathrm{\Omega }}_T}+c\mathrm{\parallel }f{\mathrm{\parallel }}_{q,{\mathrm{\Omega }}_T},c=c(N,\lambda ,q,{\mathrm{\Omega }}_T),$$ provided that $q>1+\frac{N}{2}$, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with Lipschitz boundary, $T>0$, ${\mathrm{\partial }}_p{\mathrm{\Omega }}_T$ is the parabolic boundary of ${\mathrm{\Omega }}_T$, $\omega \in L^1({\mathrm{\Omega }}_T)$ with $\omega\geq 0$,and$\lambda$isthesmallesteigenvalueofthecoefficientmatrix$A$. 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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{q,{\mathrm{\Omega }}_T}$ 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*}