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

Abstract

It is well known that a weak solution $φ$ to the initial boundary value problem for the uniformly parabolic equation $\partial_tφ-\mbox{div}(A\nabla φ) +ωφ= f $ in $Ω_T\equivΩ\times(0,T)$ satisfies the uniform estimate $$ \|φ\|_{\infty,Ω_T}\leq \|φ\|_{\infty,\partial_pΩ_T}+c\|f\|_{q,Ω_T}, \ \ \ c=c(N,λ, q, Ω_T), $$ provided that $q>1+\frac{N}{2}$, where $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, $T>0$, $\partial_pΩ_T$ is the parabolic boundary of $Ω_T$, $ω\in L^1(Ω_T)$ with $ω\geq 0$, and $λ$ is the smallest eigenvalue of the coefficient matrix $A$. This estimate is sharp in the sense that it generally fails if $q=1+\frac{N}{2}$. In this paper we show that the linear growth of this upper bound in $\|f\|_{q,Ω_T}$ can be improved. To be precise, we establish \begin{equation*} \|φ\|_{\infty,Ω_T}\leq \|φ_0\|_{\infty,\partial_pΩ_T}+c\|f\|_{1+\frac{N}{2},Ω_T}\left(\ln(\|f\|_{q,Ω_T}+1)+1\right). \end{equation*}