Logarithmic Upper Bound for Solutions of Degenerate Parabolic Equation

Abstract

In this note, we consider the following degenerate parabolic equation studied in [F. Chiarenza and R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. 137 (1984)] i.e.,

where $f=(f^1,···,f^n)$ and $Ω$ is a bounded domain in $\mathbb{R}^n$ with Lipschitz boundary, $n≥2$ and $T>0.$ In this paper, we apply Moser iteration argument to build up the explicit relationship among the coefficients $a_{i,j}(x,t)$, $f$ and the maximum norm of the solution. Meanwhile, we also find that the weighed Lebesgue space $L^{2l/(l−1)}$ to which $f$ belongs is essentially sharp in order to establish local boundedness of the solution. Here the definition of $l$ is found in Lemma 2.3. Our results cover the well-known results.