Stability, Instability, and Bifurcation in Electrified Thin Films

Abstract

In this paper, we consider an electrified thin film equation with periodic boundary conditions. When an applied voltage is sufficiently small after a finite time, we prove the global existence of unique solutions around positive constant steady states and study the asymptotic behavior of the solutions. On the other hand, when the applied voltage is constant and sufficiently large, we prove that the solutions around the constant steady states are unstable. Moreover, we prove the existence of infinitely many curves of nontrivial steady states of the electrified thin film equation around positive constant solutions at certain positive values of the voltage. Finally, as the applied voltage passes through the first bifurcation value, we obtain a unique global-in-time solution with an initially perturbed domain around nontrivial steady states which come from the first bifurcation curve, and we show that the solutions exponentially converge to the nontrivial steady-state solutions as time goes to infinity.