Exponential stability for the damped wave equation.

Abstract

In this work, we have studied the exponential stability of the damped wave equation, treated the linear case with both a global and local damping, we used a perturbed energy functional to treat the rst case and we used the multiplier method to treat the second and we gave su cient geometrical conditions on the damping domain for the stability to hold. Then, we brie y redid the main steps of the proof in the one dimensional case and as expected, it was signi cantly simpli ed (for instance, the use of only three multipliers instead of four). After that, we moved to the nonlinear case, we treated the equation with the global damping using the method introduced in [8], and then we tried to generalize it to the case of a localized damping, and there, we found some di culties treating one of the terms, so up until now, what we concluded by looking at the problem is that the result will be obtained once we get to estimate the problematic term.