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

Abstract

We consider the classical model for small-strain thermoviscoelastic evolution in isotropic Kelvin-Voigt materials with inertia, as given by $$\{\begin{array}{l}{u}_{tt}=\mu \Delta u_t+(\lambda +\mu )\nabla \nabla \cdot u_t+\hat{\mu }\Delta u+(\hat{\lambda }+\hat{\mu })\nabla \nabla \cdot u-B\nabla \Theta ,\\ \kappa {\Theta }_t=D\Delta \Theta +\mu |\nabla u_t{|}^2+(\lambda +\mu )|\mathrm{div}{u}_t{|}^2-B\Theta \mathrm{div}{u}_t.\end{array}$$ Recently, Winkler (arXiv:2602.05964) established global-in-time solvability for an initial value problem in smoothly bounded multi-dimensional domains $\Omega$ under homogeneous Dirichlet boundary conditions for the displacement variable $u$, and homogeneous Neumann boundary conditions for the temperature $\Theta$, allowing the heat capacity $\kappa$ to depend on $\Theta$. Moreover, it was shown there that the corresponding temperature is positive a.e. in $\Omega \times (0,\infty )$. In the present work, we strengthen this positivity result by deriving a quantitative lower bound for $\Theta$. 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 $\Theta (x,t)\ge c(\tau ,T)$ for a.e. $(x,t)\in \Omega \times (\tau ,T)$.

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