Benilan, Philippe; Gariepy, Ronald Strong solutions in \(L^ 1\) of degenerate parabolic equations. (English) Zbl 0828.35050 J. Differ. Equations 119, No. 2, 473-502 (1995). The paper deals with the degenerate parabolic problem of the form \(u_t = \Delta \varphi (u) + \text{div} F(u)\) on \(Q = (0,T) \times \Omega\), \(u(0, \cdot) = u_0\) on \(\Omega\), where \(\Omega \subset \mathbb{R}^N\), \(\varphi \in C^1 (\mathbb{R})\), \(F \in C^1 (\mathbb{R})^N\), \(u_0 \in L^\infty (\mathbb{R}^N)\). Assuming that \(\varphi' > 0\) a.e. on \(\mathbb{R}\) and \(|F' |^2 \leq \sigma \varphi'\) for some \(\sigma \in C (\mathbb{R})\), the authors prove that the above problem has a strong solution \(u\), i.e. \(u_t\), \(\Delta \varphi (u)\), \(\text{div} F(u)\) are functions in \(L^1_{\text{loc}} (Q)\). The proof is based on the theory of \(BV\) functions in several variables and geometric measures. Reviewer: O.Titow (Berlin) Cited in 21 Documents MSC: 35K10 Second-order parabolic equations 35K65 Degenerate parabolic equations Keywords:\(BV\) functions in several variables; geometric measures PDFBibTeX XMLCite \textit{P. Benilan} and \textit{R. Gariepy}, J. Differ. Equations 119, No. 2, 473--502 (1995; Zbl 0828.35050) Full Text: DOI