Large time behaviour of solutions for a nonlinear thermoelastic system (Q1860393)

The author is interested in the following coupled system: \[ \begin{cases} u'' - h\Delta u''+\Delta^2u-M(\|\nabla u\|^2)\Delta u+\alpha\Delta\theta = 0 & \text{in }\mathbb{R}^n\times \mathbb{R}_+,\\ \theta'-\beta\Delta\theta-\alpha\Delta u'=0 & \text{in }\mathbb{R}^n\times \mathbb{R}_+,\\u(x,0)=u_0(x),\;u'(x,0)=u_1(x),\;\theta(x,0)=\theta_0(x) & \text{in }\mathbb{R}^n,\end{cases}\tag{1} \] where \(\alpha>0\), \(\beta>0\), \(h>0\) are real constants and \(M(\cdot)\in C^1(\mathbb{R}_+)\) satisfies \(M(x)\geq 0\) for any \(x\geq 0\). By \(u\) and \(\theta\) are denoted the vertical deflection and the temperature, respectively. Using the semigroup theory and the Fourier splitting method the author establishes well-posedness and gives the decay rate of the energy corresponding to (1) as \(t\to+\infty\).