Uniform boundedness for weak solutions of quasilinear parabolic equations

Abstract

In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form \[ u t − div ⁡ A ( x , t , ∇ u ) = 0 , u_t - \operatorname {div} \mathcal {A}(x,t,\nabla u) = 0, \] where the nonlinearity A ( x , t , ∇ u ) \mathcal {A}(x,t,\nabla u) is modelled after the well-studied p 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 2 N N + 2 > p > 2 \frac {2N}{N+2}>p>2 , p = 2 p=2 , or 2 > p 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 \frac {1}{p-2} or 1 2 − p \frac {1}{2-p} , which blows up as p → 2 p \rightarrow 2 . In this note, we prove the boundedness of weak solutions in the full range 2 N N + 2 > p > ∞ \frac {2N}{N+2} > p > \infty without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of 2 N N + 1 > p > ∞ \frac {2N}{N+1} > p > \infty , we also prove an improved boundedness estimate.