Some decay properties in extensible thermoelastic beam with Gurtin–Pipkin’s law

In this article, we derive the equations that constitute the mathematical model of extensible thermoelastic beam in the context of second sound model which turns to the Gurtin–Pipkin’s one. These nonlinear governing equations are derived according to the von Kármán theory and simplified by the Euler–Bernoulli approximation in the context of Gurtin–Pipkin’s law. Even more so, the case of Fourier’s law can be recovered from the derived system one through a proper singular limit procedure, where the memory kernel collapses into the Dirac mass at zero. Then, we show that the Gurtin–Pipkin’s derived model is globally well-posed using the semigroup theory and the corresponding solutions decay exponentially under a condition on physical coefficients of the model. Finally, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we prove that the linear (without extensibility) associated semigroup is not analytic. Finally, we show that Cattaneo’s law is a particular example of Gurtin–Pipkin’s law for suitable choice of the memory kernel.