Large time existence in a thermoviscoelastic evolution problem with mildly temperature-dependent parameters
Felix Meyer
TL;DR
This work addresses the large-time existence of classical solutions for a 1D thermoviscoelastic evolution problem with mildly temperature-dependent parameters in a Kelvin-Voigt framework. It leverages a parabolic reformulation via the substitution $v:=u_t+au$ and sharp energy estimates that exploit sublinear temperature dependencies, notably the bound $|F(s)|\le C_F(1+s)^\alpha$ with $\alpha\in(0,1)$, along with small derivatives $\|\gamma'\|_{L^{\infty}}$ and $\|f'\|_{L^{\infty}}$. The main result shows that, for any prescribed time horizon $T_\star>0$, there exists a threshold $\delta_\star>0$ (depending on the data and $T_\star$) such that the maximal existence time $T_{max}$ of the classical solution satisfies $T_{max}\ge T_\star$ when the parameters stay within fixed bounds and the derivatives are sufficiently small. The approach combines energy methods, Poincaré and Gagliardo–Nirenberg inequalities, and an ODE comparison argument to rule out finite-time blow-up before $T_\star$, thereby extending local solutions to large times under mild temperature-driven variations. This provides mathematical justification for stable, long-time behavior of thermo-viscoelastic materials with temperature-dependent parameters in one spatial dimension.
Abstract
We consider \begin{align*} \label{HS} \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x +(f(Θ))_x, \\[1mm] Θ_t = DΘ_{xx} + Γ(Θ) u_{xt}^2 + F(Θ) u_{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $Θ$ in a bounded interval $Ω\subset\mathbb{R}$. \abs This model is a generalization of the classical system for the description of strain and temperature evolution in a thermo-viscoelastic material following a Kelvin-Voigt material law, in which $γ\equiv Γ$ and $f\equiv F$. Different variations of this model have already been analyzed in the past and the present study draws upon a known result concerning the existence of classical solutions, which are local in time, for suitably smooth initial data, arbitrary $a>0$, $D>0$ and $γ,f\in C^2([0,\infty))$ as well as $Γ,F\in C^1([0,\infty))$ with $γ>0,Γ\ge0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of $γ$ and $f$, and further $|F(s)|\le C_F(1+s)^α$ for some $C_F>0$ and $α\in(0,1)$. In particular, for any given $T_\star$, initial mass $M$ and $0<\underlineγ<\overlineγ$, there exists a constant $δ_\star(M,T_\star,a,D, Ω, \underlineγ, \overlineγ,C_F,α)>0$, such that if $$\underlineγ\leγ\le \overlineγ\quad\mbox{ and }\quad 0\le Γ\le \overlineγ\quad \mbox{ as well as } \quad\|γ'\|_{L^\infty([0,\infty))}\le δ_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le δ_\star $$ hold, the maximal existence time of the classical solution to $(\star)$ surpasses $T_\star$.