# Weak solutions to gradient flows of functionals with inhomogeneous growth in metric spaces

## TL;DR

This paper introduces a new approach to define and analyze weak solutions for gradient flows in complex mathematical spaces.

## Contribution

The paper introduces a novel framework for weak solutions in metric spaces with inhomogeneous growth functionals.

## Key findings

- A new definition of weak solutions for gradient flows is established.
- Existence and uniqueness of these solutions are proven.
- The approach is applied to functionals with inhomogeneous growth.

## Abstract

We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solution to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.

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Source: https://tomesphere.com/paper/PMC12050237