Exponential stability of a thermoelastic nonlinear shear beam

In this paper, we study the stabilization of a thermoelastic nonlinear shear beam model. We incorporate thermal dissipation into the transverse displacement equation, following Fourier theory. The Shear beam model constitutes an improvement over the Euler-Bernoulli beam model by adding the shear distortion effect but without rotary inertia. Unlike Euler-Bernoulli and Rayleigh beam models, the Shear model has two dependent variables for dynamic of the beam. First, by using the Faedo-Galerkin method, we prove the well-posedness of the system. Second, by using the integral-type multiplier method, we prove that the energy of the system decays exponentially regardless of any relationship between coefficients of the system, since the system has only one wave speed. Numerically, by using a finite element scheme in space and both implicit Euler and Crank-Nicolson methods in time, we prove that the associated discrete energy decays. Then, we present a priori error estimates from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, we present some numerical experiments, to support our theoretical results.