Energy Decay for a Timoshenko–Boltzmann System with Memory and Gurtin–Pipkin Thermal Law

ABSTRACT

This paper establishes the well‐posedness and stability properties of a coupled thermo‐viscoelastic Timoshenko–Boltzmann system featuring two nonlocal mechanisms: (i) shear deformation with Boltzmann‐type viscoelastic memory, and (ii) heat conduction governed by the Gurtin–Pipkin nonlocal thermal law. Through semigroup operator theory, we first prove the existence of a unique solution flow in an appropriate history‐weighted energy space. Subsequently, using energy methods with memory‐dependent functional constructions, we characterize the energy decay rates in terms of the relaxation kernel's dissipation strength. When the viscoelastic kernel decays exponentially, the system exhibits exponential stabilization. For weakly decaying kernels, we establish the polynomial decay rate. These results quantify how the interplay between viscoelastic memory dissipation and nonlocal thermal damping governs the system's stabilization dynamics.