Improved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space (Q1868788)

Better decay estimates of solutions to the Cauchy problem for the linear equation \(u_{tt}-u_{xx}+u_{t}=0\) with restricted initial data are obtained by using the method suggested in the paper of \textit{R. Ikehata} and \textit{T. Matsuyama} [Sci. Math. Jpn. 55, 33-42 (2001; Zbl 1003.35020)]. The results are applied to prove global existence of solutions to the semilinear equation \(u_{tt}-u_{xx}+u_{t}=|u|^{p-1}u\) for small initial data.