Uniform boundedness for weak solutions of quasilinear parabolic equations.

In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form ut − div A(x,t,∇ u) = 0, where the nonlinearity A(x,t,∇ u) is modelled after the well-studied p-Laplace operator. The question of boundedness has received a lot of attention over the past several decades with the existing literature showing that weak solutions in either 2N/N+2 < p < 2, p = 2, or 2 < p, are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included an exponent of the form 1/p−2 or 1/2−p, which blows up as p → 2. In this note, we prove the boundedness of weak solutions in the full range 2N/N+2 < p < ∞ without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of 2N/N+1 < p < ∞, we also prove an improved boundedness estimate. [ABSTRACT FROM AUTHOR]