# Well-posedness of a Model for the Growth of Tree Stems and Vines

## TL;DR

This paper proves the well-posedness of a PDE model describing the growth of tree stems and vines, accounting for cell elongation, gravity, obstacles, and unilateral constraints, until a breakdown occurs.

## Contribution

It establishes the existence and uniqueness of solutions for the growth model and characterizes the reaction forces via energy minimization under constraints.

## Key findings

- The model is well-posed until a breakdown configuration.
- Reaction forces are characterized by energy minimization.
- The PDE includes state constraints representing obstacles.

## Abstract

The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles. The main theorem shows that the evolution problem is well posed, until a specific "breakdown configuration" is reached. A formula is proved, characterizing the reaction produced by unilateral constraints. At a.e. time t, this is determined by the minimization of an elastic energy functional under suitable constraints.