Finite-time blow-up for a nonlinear thermoelastic Kirchhoff system

We study a scalar thermoelastic system whose mechanical part contains a nonlocal Kirchhoff term, linear viscous damping, and a superlinear source, with p > 4 expressing that the destabilizing source grows faster than the nonlocal stiffening energy. For dissipative energy solutions with the complete mechanical-thermal damping form uniformly positive, we show that if 4 < p <= 2* and the initial energy is negative, every such solution has finite lifespan: an auxiliary functional satisfies a superlinear differential inequality giving an explicit upper bound for the existence time.