Fast energy decay for damped wave equations with a potential and rotational inertia terms

TL;DR

This paper investigates the energy decay of damped wave equations with potential and rotational inertia in one dimension, demonstrating fast decay rates despite the lack of traditional inequalities.

Contribution

It provides new results on energy decay for a one-dimensional damped wave model with potential and inertia, overcoming challenges due to the absence of Hardy and Poincaré inequalities.

Findings

Proves fast energy decay in one-dimensional damped wave equations

Establishes L^2-decay of solutions over time

Highlights the role of potential in compensating for analytical difficulties

Abstract

We consider damped wave equations with a potential and rotational inertia terms. We study the Cauchy problem for this model in the one dimensional Euclidean space and we obtain fast energy decay and L^2-decay of the solution itself as time goes to infinity. Since we are considering this problem in the one dimensional space, we have no useful tools such as the Hardy and/or Poincar\'e inequalities. This causes significant difficulties to derive the decay property of the solution and the energy. A potential term will play a role for compensating these weak points.