Well-Posedness and decay analysis for a thermoelastic wave system with spatially variable-exponent damping and time-varying delay

Abstract

We study a thermoelastic wave–heat system on a smooth bounded domain in which the mechanical velocity is damped both by a spatially variable-exponent power and by a time-varying delay control. The exponent field s(·) is log-Hölder with 2 ≤ s₋ ≤ s(x) ≤ s₊, where s₊ lies strictly below a scaling-critical bound; the delay is uniformly positive with derivative < 1; the delayed coefficient may change sign but is small relative to the primary damping profile. Encoding the delayed velocity through a first-order transport equation, we construct a Lyapunov functional that controls the elastic, thermal, and delay contributions and yields coercive dissipation. Using a Faedo–Galerkin approximation, compactness arguments in variable-exponent spaces, and the Minty–Browder method, we obtain global existence and uniqueness of strong solutions for admissible data. The same functional, together with a Komornik-type integral inequality, yields an explicit exponential decay rate determined by the time integral of the primary damping profile. The approach also adapts to Cattaneo heat conduction and to distributed delays.

Keywords

thermoelasticity / variable-exponent damping / time-varying delay control / global strong well-posedness / exponential energy decay / Lyapunov–Komornik framework / Faedo–Galerkin scheme / Minty–Browder monotonicity

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