Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes

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