Local well-posedness, potential-well stability, and negative-energy blow-up for a semilinear thermoelastic system with variable coefficients

We study a scalar thermoelastic wave-heat system with a superquadratic source, linear viscous damping, a time-dependent elastic tensor, and a nonlinear bulk heat sink. For a source exponent in a natural L^2-subcritical range we prove local existence, uniqueness, continuous dependence, and a continuation criterion in the energy class, and derive the exact nonautonomous energy identity. In the autonomous unforced case, the potential-well construction gives a stable set of data for which the solution is global; conversely, for nonincreasing elastic coefficients and nondecreasing source coefficient, every maximal solution with negative initial energy has finite lifespan.