Equations with Lipschitz nonlinearities

Abstract

We start here by studying the porous media equation problem (1.1) when β: ℝ → ℝ is monotonically increasing and Lipschitz continuous. The main reason is that general maximal monotone graphs β can be approximated by their Yosida approximations βϵ which are Lipschitz continuous and monotonically increasing. So, several estimates proved in this chapter will be exploited later for studying problems with more general β. We also note that there are significant physical problems with β Lipschitzcontinuous as in the case of the Stefan two phase problem presented in Sect. 1.1.1.