Decay Estimate for the Time-Delayed Fourth-Order Wave Equations

Abstract

The objective of this article is to analyze the stability of solutions for the following fourth- order nonlinear wave equations with an internal delay term: \begin{equation*} u_{tt} + \Delta^2 u + u + \sigma_1(t) |u_{t}(x,t)|^{2m-2} u_t(x,t) + \sigma_2(t) |u_{t} (x,t-\tau)|^{2m-2} u_t(x,t-\tau) = 0. \end{equation*} We obtain appropriate conditions on $\sigma_1(t)$ and $\sigma_2(t)$ for the decay properties of the solutions. The multiplier technique and nonlinear integral inequalities are used in the proof.