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 $$u_t - \text{div} \mathcal{A}(x,t,\nabla u) = 0,$$ where the nonlinearity $\mathcal{A}(x,t,\nabla u)$ is modelled after the well studied $p$-Laplace operator. The question of boundedness has received lot of attention over the past several decades with the existing literature showing that weak solutions in either $\frac{2N}{N+2}<p<2$, $p=2$ 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 $\frac{1}{p-2}$ or $\frac{1}{2-p}$ which blows up as $p \rightarrow 2$. In this note, we prove the boundedness of weak solutions in the full range $\frac{2N}{N+2} < p < \infty$ without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of $\frac{2N}{N+1} < p < \infty$, we also prove an improved boundedness estimate.