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 $$\begin{cases} u_{tt} = μΔu_t + (λ+μ)\nabla\nabla\cdot u_t + \hatμΔu + (\hatλ+\hatμ)\nabla\nabla\cdot u - B\nablaΘ, \\ κΘ_t = DΔΘ+ μ|\nabla u_t|^2 + (λ+μ)|\mathrm{div}\,u_t|^2 - BΘ\,\mathrm{div}\,u_t. \end{cases}$$ Recently, Winkler (arXiv:2602.05964) established global-in-time solvability for an initial value problem in smoothly bounded multi-dimensional domains $Ω$ under homogeneous Dirichlet boundary conditions for the displacement variable $u$, and homogeneous Neumann boundary conditions for the temperature $Θ$, allowing the heat capacity $κ$ to depend on $Θ$. Moreover, it was shown there that the corresponding temperature is positive a.e. in $Ω\times(0,\infty)$. In the present work, we strengthen this positivity result by deriving a quantitative lower bound for $Θ$. Our analysis applies to heat capacities $κ$ with power-law scaling for small arguments, precisely: either $κ(ξ)\simξ^α$ for some $α\ge 0$, or $κ(ξ)\geξ$, in a suitable neighborhood of $ξ=0$. Under these structural assumptions on $κ$, we show that for every $τ>0$ and $T>τ$ there exists $c(τ,T)>0$ such that $Θ(x,t)\ge c(τ,T)$ for a.e. $(x,t)\inΩ\times(τ,T)$.