Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes

TL;DR

This paper investigates the long-term dynamics of a degenerate quasilinear hyperbolic system modeling elastic membranes, revealing exponential convergence with damping and extended lifespan without damping, using advanced Nash-Moser-Hörmander techniques.

Contribution

It introduces new results on the asymptotic behavior of elastic membranes governed by a complex hyperbolic system, applying a novel Nash-Moser-Hörmander theorem.

Findings

Exponential convergence of perturbed spheres with damping.

Extended lifespan of solutions without damping.

Application of a new Nash-Moser-Hörmander theorem.

Abstract

The paper studies the long time behavior of a system that describes the motion of a piece of elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, and also includes a damping term that models the dissipative nature of genuine physical systems. With the presence of damping, a small perturbation of the sphere converges exponentially in time to the sphere, and without the damping the evolution that is $\varepsilon$-close to the sphere has life span longer than $\varepsilon^{-1/6}$. Both results are proved using a new Nash-Moser-H\"{o}rmander type theorem proved by Baldi and Haus.