WIAS Preprint No. 1226, (2007)

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



Authors

  • Rossi, Riccarda
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Savaré, Giuseppe

2010 Mathematics Subject Classification

  • 35K55 49Q20 58E99

Keywords

  • Doubly nonlinear equations, analysis in metric spaces, existence and approximation results

Abstract

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 abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.

Appeared in

  • Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), VII (2008) pp. 97--169.

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