Chapter 3 Current issues on singular and degenerate evolution equations

Abstract

Publisher Summary This chapter discusses the current issues on singular and degenerate evolution equations. It discusses the question of the regularity of the weak solutions of singular and degenerate quasilinear parabolic equations, proving their Holder character. The chapter presents a precise definition of weak solution and the derivation of the building blocks of the theory: the local energy and logarithmic estimates. The chapter also presents the classical approach of De Giorgi to uniformly elliptic equations. The chapter reviews the classical results concerning Harnack inequalities, along with providing a proof of the Harnack inequality both in the degenerate and singular case. The chapter shows that for positive solutions of the singular p -Laplace equation, an “elliptic” Harnack inequality holds. The phenomenon of the extinction of the solution in finite time is discussed in the chapter. It also discusses physical motivations concerning Stefan-like equations and demonstrates, through the Kruzkov–Sukorjanski transformation, the deep links between degenerate equations and Stefan-like equations.