Smooth Dynamics of a New Semilinear Bresse–Fourier System: A Semigroup Approach

This paper is concerned with a thermoelastic Bresse system that models the dynamics of arched beams. It analyzes the effect of temperature on each constitutive equation, namely the shear force, bending moment, and axial force. Within a framework that uses Fourier's law of heat conduction, the exponential stability of the model is studied. To make the analysis more involved, we consider Dirichlet boundary conditions, which introduce technical challenges in handling the boundary terms. These challenges are addressed using a truncation argument and an observability inequality. In addition, we establish the existence of a smooth global attractor when the system is submitted to a nonlinear foundation. We do not use energy methods. Instead, we explore the uniform stability of the homogeneous problem.