Hieber, Matthias; Rehberg, Joachim Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains. (English) Zbl 1221.35194 SIAM J. Math. Anal. 40, No. 1, 292-305 (2008). The authors deal with quasilinear systems of reaction-diffusion equations with mixed Dirichlet-Neumann boundary conditions with non regular coefficients. They prove the existence of a solution using the powerful machinery of maximal \(L^p\) estimates. Indeed this approach is very elegant with respect to the usual ones based on the construction of an appropriate evolution operator. On the other hand, this compels the authors to restrict their analysis to the case of dimensions two and three, as they need some embedding properties, that do not hold in higher dimensions (see Shamir’s counterexample). Reviewer: Vincenzo Vespri (Firenze) Cited in 24 Documents MSC: 35K55 Nonlinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35R05 PDEs with low regular coefficients and/or low regular data Keywords:mixed Dirichlet-Neumann boundary conditions; maximal \(L^p\) regularity; \(L^\infty\)-coefficients; non regular coefficients PDFBibTeX XMLCite \textit{M. Hieber} and \textit{J. Rehberg}, SIAM J. Math. Anal. 40, No. 1, 292--305 (2008; Zbl 1221.35194) Full Text: DOI