Existence of periodic solutions for a nonlinear plate coupling system with thermal memory and external forces

This paper is devoted to investigating the existence and uniqueness of T-periodic solutions for a nonlinear thermoelastic plate coupling system with thermal memory effects and time-periodic external forces, derived from the non-Fourier heat flux law—a model more physically realistic for characterizing the thermal response of materials with transient heat conduction behavior. To address the mathematical challenges of this coupled system, we first transform the original high-order system into an equivalent first-order evolution system via auxiliary and memory variable substitutions. Using the Galerkin method to construct finite-dimensional approximate solutions, we then apply the Leray-Schauder fixed-point theorem to prove the existence of approximate periodic solutions, deriving uniform a priori estimates for their derivatives in Hilbert space V 0 via Hölder’s, Poincaré’s, and Gronwall’s inequalities. The Sobolev compact embedding theorem verifies the convergence of approximate solutions, establishing the existence of T-periodic solutions for the original system; uniqueness is further proven via an energy difference functional and Gronwall’s lemma under a smallness condition on external forces. This work enriches the theoretical framework for periodic solutions of memory-type thermoelastic coupling systems and provides a foundation for engineering dynamic analysis of plate structures.