Rates of decay of a nonlinear model in thermoelasticity (Q2712992)

A coupled system of equations in the nonlinear thermoelasticity consisting of a nonlinear hyperbolic equation with a heat type equation in the whole line is considered. The existence and uniqueness of the global (weak) solution has been proved by using semigroup theory. Some estiamte of the solutions via the Fourier transform together with a choice of a convenient Lyapunov function has been obtained. The asymptotic behavior of the total energy as \(t\to+\infty\) is investigated. The proof is a similar to the technique introduced by \textit{M. E. Schonbek} [Commun. Partial Differ. Equations 11, 733-763 (1986; Zbl 0607.35071)] for analysing similar problem for Navier-Stokes equations.