Fatou theorem for the solutions of some nonlinear equations
The paper deals with the equation (1) \(u_ t = \Delta \Phi (u)\) where \(\Phi\) is continuous, strictly increasing in the interval \([0, \infty)\) and satisfies certain growth conditions, such that \(\Phi (0) = 0\), \(\Phi (u) > 0\) for \(u > 0\). Equation (1) generalizes the porous medium and fast diffusions equations. Assuming \(u(x,t)\) to be a nonnegative continuous weak solution of (1) \((x \in \mathbb{R}^ n\), \(0 < t < T)\) the author proves several theorems concerning the limit of \(u(\cdot,t)\) when \(t \to 0 +\). The paper generalizes the results concerning the porous medium equation given in [\textit{B. E. J. Dahlberg}, \textit{E. B. Fabes} and \textit{C. E. Kenig}, Proc. Am. Math. Soc. 91, 205-212 (1984; Zbl 0552.35047)].
scientific article; zbMATH DE number 599933