Uniform Stability of One‐Dimensional Wave Equations With Arbitrary Local Viscoelastic Damping

ABSTRACT In this paper, we discuss the one‐dimensional wave equations with local viscoelastic damping. It is worth pointing out that there is no restriction on the viscoelastic damping term in the system, and there are no other types of damping terms. Although it is only a one‐dimensional system, this kind of local damping problem has not been studied in the previous literature. In previous studies, most of them discussed the global damping or local viscoelastic damping which must contain an end point. Under such geometric constraints, the previous multiplier method which used to estimate energy cannot be directly applied, as detailed in the paper. For the problem, we make some elaborate techniques, and we obtain the decay rate estimate of the system energy under the assumption of a weaker relaxation function. Thus, we generalize and improve the previous results under the weaker condition.