Gevrey Class for Locally Three-Phase-Lag Thermoelastic Beam System

In this article we study the behavior of the solutions for the three-phase-lag heat equation with localized dissipation on an Euler–Bernoulli beam model. We show that semigroup S(t) associated with the problem is of Gevrey class 5 for \(t>0\) $t>0$ . If the coefficients satisfy \(\tau _\alpha > k^{*}\tau _q\) ${\tau }_{\alpha }>k^{\ast }{\tau }_q$ , the solutions are always exponentially stable.