Comparison results for nonlinear parabolic equations with monotone principal part (Q5945766)

An abstract comparison principle for solutions of quasilinear parabolic problems with monotone principal part is proved at first. This result is later applied to a parabolic problem having \(p\)-Laplacian as its principal part: NEWLINE\[NEWLINE{{du}\over {dt}}-\Delta_p u = B u,NEWLINE\]NEWLINE where \(p>1\), \(\Delta_p u:= \text{div} (|\nabla u|^{p-2}\nabla u)\) and \(B\) is a single-valued operator satisfying certain additional conditions (\(B\) would be nonmonotone and even non-Lipschitz). At the end of the paper more applications of the abstract result are shown. Let us note that the abstract comparison principle is an extension of that known for semilinear parabolic equations (i.e. principal part is linear) presented in \textit{J. M. Arrieta, A. N. Carvalho} and \textit{A. Rodriguez-Bernal} [Commun. Partial Differ. Equations 25, 1-37 (2000; Zbl 0953.35021)].