A metric approach to a class of doubly nonlinear evolution equations and applications

Riccarda Rossi, Alexander Mielke, Giuseppe Savaré
2009 Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V  
This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5] ), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to
more » ... t doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in L 1 spaces. (2000) : 35K55 (primary); 49Q20, 58E99 (secondary). Mathematics Subject Classification Hence, supposing that the functional E : X → (−∞, +∞] complies with the chain rule (1.10), we say that an absolutely continuous curve u : (0, T ) → X satisfies the metric formulation of (DNE) if the map t ∈ (0, T ) → E(u(t)) is absolutely continuous and d dt E(u(t)) ≤ −ψ |u |(t) − ψ * |∂E| (u(t)) for a.e. t ∈ (0, T ). (1.12)
doi:10.2422/2036-2145.2008.1.04 fatcat:us27dkdylbeuxhfuprkfirzdge