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

Abstract

It is well known that a weak solution $\phi$ to the initial boundary value problem for the uniformly parabolic equation $\partial_tφ-\mbox{div}(A\nabla φ) +ωφ= f $ in ${\Omega }_T\equiv \Omega \times (0,T)$ satisfies the uniform estimate $$\Vert \phi {\Vert }_{\infty ,{\Omega }_T}\le \Vert \phi {\Vert }_{\infty ,{\partial }_p{\Omega }_T}+c\Vert f{\Vert }_{q,{\Omega }_T},c=c(N,\lambda ,q,{\Omega }_T),$$ 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 $\omega \ge 0$, 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 $\Vert f{\Vert }_{q,{\Omega }_T}$ 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*}