Minimal solutions to generalized Lambda-semiflows and gradient flows in metric spaces

TL;DR

This paper introduces the concept of minimal solutions within generalized Lambda-semiflows in metric spaces, proving their uniqueness and their role in generating all solutions through reparametrization, with applications to gradient flows.

Contribution

It defines and analyzes minimal solutions in generalized Lambda-semiflows, establishing their uniqueness and applicability to gradient flows in metric spaces.

Findings

Minimal solutions are unique for given ranges.

Minimal solutions generate all other solutions via reparametrization.

Applications to gradient flows in metric spaces are demonstrated.

Abstract

Generalized Lambda-semiflows are an abstraction of semiflows with non-periodic solutions, for which there may be more than one solution corresponding to given initial data. A select class of solutions to generalized Lambda-semiflows is introduced. It is proved that such minimal solutions are unique corresponding to given ranges and generate all other solutions by time reparametrization. Special qualities of minimal solutions are shown. The concept of minimal solutions is applied to gradient flows in metric spaces and generalized semiflows. Generalized semiflows have been introduced by Ball.