Global smooth solutions in a one-dimensional thermoviscoelastic model with temperature-dependent paramaters
Felix Meyer
TL;DR
This work analyzes a one-dimensional thermoviscoelastic Kelvin–Voigt type model with temperature-dependent parameters $\gamma(\Theta)$ and $f(\Theta)$. The authors establish global existence of classical solutions for arbitrarily large smooth initial data under assumptions that include $c_\gamma<\gamma(\zeta)<C_\gamma$, $\gamma''(\zeta)\le0$, $f(0)=0$, $|f'|\le C_f$, and sublinear growth $|f(\zeta)|\le C_f(1+\zeta)^\alpha$ with $0<\alpha<5/6$. The main approach combines energy estimates, a Moser iteration to obtain $L^{\infty}$ bounds for $u_t$, and a Porzio–Vespri Hölder regularity step to control solution smoothness, ultimately showing global solvability in 1D by an extensibility criterion. These results provide mathematical validation for 1D thermo-viscoelastic models with temperature-dependent coefficients and may guide extensions to more general settings. The work leverages 1D Sobolev embeddings and interpolation inequalities to close the estimates.
Abstract
This manuscript is concerned with the system \begin{align*} \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + (a(x,t) u_x)_x +(f(Θ))_x, \\[1mm] Θ_t = DΘ_{xx} + γ(Θ) u_{xt}^2 + f(Θ) u_{xt}, \end{array} \right. \end{align*} which is used to describe thermoviscoelastic developments in one-dimensional Kelvin-Voigt materials. \abs It is assumed that $a,γ$ and $f$ are sufficiently smooth functions that satisfy $$c_γ<γ(ζ)<C_γ, \quad γ''(ζ) \le 0,\quad f(0)=0, \quad |f'(ζ)|\le C_f \quad \mbox{ and } |f(ζ)|\le C_f(1+ζ)^α\quad \mbox{ for all }ζ\ge 0 $$ and some positive constants $c_γ,C_γ,C_f>0$ and $α\in (0,5/6)$. Under these conditions, this study then establishes a result on the existence of global classical solutions for sufficiently smooth but arbitrarily large initial data.