New perturbation methods for nonlinear parabolic problems (Q456614)

Regularity results for nonlinear degenerate parabolic systems of the type NEWLINE\[NEWLINE{\partial u\over\partial t}- \text{div}(\gamma(x,t) a(Du))= -\text{div}\,G(x,t)NEWLINE\]NEWLINE are derived through a local perturbation procedure (\(a(Du)\) is regarded as a perturbation of the operator \(|Du|^{p-2}Du\)). Delicate parabolic perturbation techniques are designed by the authors. In particular, the authors obtain the following kind of results:NEWLINENEWLINE -- Local boundedness of \(|Du|\) for systems that are parabolic only for \(|Du|\approx\infty\).NEWLINENEWLINE -- Continuity of \(|Du|\), when the space variable coefficients are Dini continuous.NEWLINENEWLINE -- Hölder continuity of \(|Du|\), when the coefficients are Hölder continuous.NEWLINENEWLINE The authors extend some recent nonlinear Schauder estimates of \textit{M. Misawa} [Manuscr. Math. 109, No. 4, 419--454 (2002; Zbl 1026.35025)].