Quantitative positivity properties of the temperature in multi-dimensional thermoviscoelastic systems

We consider the classical model for small-strain thermoviscoelastic evolution in isotropic Kelvin-Voigt materials with inertia, as given by $$\{\begin{array}{l}{u}_{tt}=\mu \mathrm{\Delta }{u}_t+(\lambda +\mu )\mathrm{\nabla }\mathrm{\nabla }\cdot u_t+\widehat{\mu }\mathrm{\Delta }u+(\widehat{\lambda }+\widehat{\mu })\mathrm{\nabla }\mathrm{\nabla }\cdot u-B\mathrm{\nabla }\mathrm{\Theta },\\ \kappa {\mathrm{\Theta }}_t=D\mathrm{\Delta }\mathrm{\Theta }+\mu \mathrm{\mid }\mathrm{\nabla }{u}_t{\mathrm{\mid }}^2+(\lambda +\mu )\mathrm{\mid }\mathrm{d}\mathrm{i}\mathrm{v}{u}_t{\mathrm{\mid }}^2-B\mathrm{\Theta }\mathrm{d}\mathrm{i}\mathrm{v}{u}_t\mathrm{.}\end{array}$$ Recently, Winkler (arXiv:2602.05964) established global-in-time solvability for an initial value problem in smoothly bounded multi-dimensional domains $\mathrm{\Omega }$ under homogeneous Dirichlet boundary conditions for the displacement variable $u$, and homogeneous Neumann boundary conditions for the temperature $\mathrm{\Theta }$, allowing the heat capacity $\kappa$ to depend on $\mathrm{\Theta }$. Moreover, it was shown there that the corresponding temperature is positive a.e. in $\mathrm{\Omega }\times (0,\mathrm{\infty })$. In the present work, we strengthen this positivity result by deriving a quantitative lower bound for $\mathrm{\Theta }$. Our analysis applies to heat capacities $\kappa$ with power-law scaling for small arguments, precisely: either $\kappa (\xi )\sim {\xi }^{\alpha }$ for some $\alpha \ge 0$, or $\kappa (\xi )\ge \xi$, in a suitable neighborhood of $\xi =0$. Under these structural assumptions on $\kappa$, we show that for every $\tau >0$ and $T>\tau$ there exists $c(\tau ,T)>0$ such that $\mathrm{\Theta }(x,t)\ge c(\tau ,T)$ for a.e. $(x,t)\in \mathrm{\Omega }\times (\tau ,T)$.