(untitled) (Communications in Mathematical Research, 2025)
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 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.