On the $L^2 ({\mathbb {R}})$-Norm Decay Estimates for Two Cauchy Systems of Coupled Wave Equations Under Frictional Dampings

Abstract

In this paper, we consider two Cauchy systems of coupled two wave equations in the whole line ${\mathbb {R}}$ under one or two frictional dampings, where the coupling terms are either of order one with respect to the time variable or of order two with respect to the space variable. We prove some $L^2 ({\mathbb {R}})$-norm decay estimates of solutions and their higher-order derivatives with respect to the space variable, where the decay rates depend on the number of the present frictional dampings, the regularity of the initial data, and some connections between the speeds of wave propagation of the two wave equations. Both our systems are considered under weaker conditions on the coefficients than the ones considered in the literature and they include the case where only one frictional damping is present, so they generate new difficulties and represent new situations that have not been studied earlier.