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

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 Poincaré 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.