Strong solutions in \(L^ 1\) of degenerate parabolic equations
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.
scientific article; zbMATH DE number 784949
scientific article