We study stability issues for a dynamical system consisting of a wave equation and a quasilinear parabolic equation. The nonlinearity involves the p-Laplacian, and the coupling involves a fractional Laplacian with exponent \(\mu \) that lies in the interval [0, 1]. First, we revisit the corresponding linear system, which, in the three-dimensional space setting, describes the longitudinal motion of a thermoelastic material when \(\mu =1\) . For this linear model, which corresponds to \(p=2\) , we improve an earlier stability result for the associated semigroup when \(\mu \) lies in the interval [0, 1); in particular, we show that the semigroup decays with rate \(O(t^{-\frac{1}{2(1-\mu )}})\) when the time variable goes to infinity. We also prove that this decay rate is optimal. We prove those results using frequency domain tools and suitable interpolation inequalities. Next, we tackle the nonlinear system for \(\mu =1\) . After discussing the wellposedness of this system, we prove that its energy decays at a rate, which for some space dimensions, specializes to \(O(t^{-\frac{2}{p-2}})\) as t goes to infinity. For this proof, we use the perturbed energy method combined with a Gagliardo-Nirenberg interpolation inequality.
