Well‐Posedness, General Stability, and Numerical Analysis of a Nonlinear Thermodiffusive Timoshenko System with Second Sound, Infinite Hereditary Memory, Distributed Delay, and Logarithmic Dissipation

ABSTRACT In this paper, we investigate a nonlinear thermoelastic Timoshenko system involving thermodiffusion effects, second sound propagation, infinite hereditary memory, distributed delay, and nonlinear logarithmic damping. The model describes the evolution of the transverse displacement , the rotational angle , the temperature variation , the heat flux governed by Cattaneo's law, and the diffusive variable associated with thermodiffusion phenomena. The considered coupled thermo‐viscoelastic system is governed by for . The thermal component is modeled through the Cattaneo heat conduction law which introduces finite propagation speed of thermal waves and incorporates second sound effects. The viscoelastic behavior is described by an infinite memory operator of the form while the distributed delay contribution generates additional hereditary interactions depending on the past history of the rotational velocity. First, the problem is reformulated as an abstract evolution equation in a suitable Hilbert space involving weighted history spaces associated with the infinite memory term. By combining semigroup theory, dissipativity methods, variational arguments, and the Lumer–Phillips theorem, we establish the global well‐posedness of the coupled thermoelastic system. Next, suitable Lyapunov functionals are constructed in order to compensate the strong coupling generated by the hereditary memory operator, the second sound mechanism, the thermodiffusion effects, and the distributed delay contribution. The stability analysis is particularly delicate because the logarithmic damping term does not satisfy standard coercivity assumptions near the origin, while the infinite memory contribution introduces highly nonlocal effects depending on the entire past history of the solution. Using refined multiplier techniques, weighted energy estimates, and suitable compensating inequalities, we derive a general decay estimate for the total energy. More precisely, under the relaxation assumption where is a positive nonincreasing function, we prove that the total energy satisfies The obtained result provides a unified stability framework that includes exponential, polynomial, and more general decay rates as particular cases according to the asymptotic behavior of the relaxation kernel. Several numerical simulations and computational experiments are also developed through fully implicit finite difference approximations preserving the dissipative structure of the continuous model. The numerical tables and energy profiles clearly illustrate the influence of second sound effects, distributed delay, hereditary memory, and logarithmic damping on the asymptotic behavior and stabilization properties of the thermoelastic system. The computational results are shown to be in complete agreement with the theoretical analysis. The present contribution considerably extends and improves several previous works devoted to thermoelastic Timoshenko systems by simultaneously combining thermodiffusion coupling, second sound propagation, infinite hereditary memory, distributed delay, and nonlinear logarithmic dissipation within a unified analytical and numerical framework.