Stability analysis in extensible thermoelastic beam with microtemperatures

In this article we derive the equations that constitute the nonlinear mathematical model of one-dimensional extensible elastic beam with temperature and microtemperatures effects. The nonlinear governing equations are derived by applying the Hamilton principle to full von Karman equations in the framework of Euler-Bernoulli beam theory. The model takes account of the effects of extensiblity and rotational inertia where the dissipations are entirely contributed by temperature and microtemperatures. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. Then, using the multiplier method, we show that the solutions decay exponentially if (4.1) holds. Finally we consider the case of zero thermal conductivity and we show that the dissipation given only by the microtemperatures is strong enough to produce exponential stability if (4.1) holds. By an approach based on the Gearhart-Herbst-Pruss-Huang theorem, we prove that the linear (without extensiblity) associated semigroup is not analytic.