Large time existence in a thermoviscoelastic evolution problem with mildly temperature-dependent parameters

TL;DR

This paper proves that classical solutions to a generalized thermo-viscoelastic system with mildly temperature-dependent parameters can exist for arbitrarily long times, under certain boundedness and sublinearity conditions on the parameters.

Contribution

It establishes conditions under which the existence time of classical solutions can be extended arbitrarily, generalizing previous local existence results for thermo-viscoelastic models.

Findings

Existence times can be made arbitrarily large under specific parameter bounds.

Sublinear temperature dependencies of parameters are crucial for long-time existence.

The results apply to a broad class of thermo-viscoelastic systems with mildly temperature-dependent coefficients.

Abstract

We consider \begin{align*} \label{HS} \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x +(f(\Theta))_x, \\[1mm] \Theta_t = D\Theta_{xx} + \Gamma(\Theta) u_{xt}^2 + F(\Theta) u_{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $\Theta$ in a bounded interval $\Omega\subset\mathbb{R}$. \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 $\gamma\equiv \Gamma$ 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 $\gamma,f\in C^2([0,\infty))$ as well as $\Gamma,F\in C^1([0,\infty))$ with $\gamma>0,\Gamma\ge0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of $\gamma$ and $f$, and further $|F(s)|\le C_F(1+s)^\alpha$ for some $C_F>0$ and $\alpha\in(0,1)$. In particular, for any given $T_\star$, initial mass $M$ and $0<\underline\gamma<\overline\gamma$, there exists a constant $\delta_\star(M,T_\star,a,D, \Omega, \underline\gamma, \overline\gamma,C_F,\alpha)>0$, such that if $$\underline\gamma \le\gamma\le \overline\gamma\quad\mbox{ and }\quad 0\le \Gamma\le \overline\gamma \quad \mbox{ as well as } \quad\|\gamma'\|_{L^\infty([0,\infty))}\le \delta_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le \delta_\star $$ hold, the maximal existence time of the classical solution to $(\star)$ surpasses $T_\star$.