A metric approach to a class of doubly nonlinear evolution equations and applications. (English) Zbl 1183.35164

Rossi, Riccarda; Mielke, Alexander; Savaré, Giuseppe

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7, No. 1, 97-169 (2008).

The paper deals with the study of a class of doubly nonlinear evolution equations in the framework of general metric spaces. More precisely, the authors propose a suitable metric formulation, in fact adapting the notion of curve of maximal slope to the doubly nonlinear case. In this context, they prove an existence and approximation result for the related Cauchy problem, in the case of a superlinear dissipation functional, by means of an approximation scheme based on time discretization. Moreover they apply their results to metric evolutions in \(L^1\) spaces, by providing some examples with explicit calculation of the metric solution. Finally some application in reflexive Banach spaces are also given. The study has been developed in following articles of the same authors to the case of the analysis of rate independent problems.

Reviewer: Elvira Mascolo (Firenze)

MSC:

35K55	Nonlinear parabolic equations
49Q20	Variational problems in a geometric measure-theoretic setting

Keywords:

existence; approximation result; Cauchy problem; superlinear dissipation functional; time discretization