Large-data solutions in multi-dimensional thermoviscoelasticity with temperature-dependent viscosities

TL;DR

This paper proves the global existence of weak solutions for a complex multi-dimensional thermoviscoelastic system with temperature-dependent viscosities, modeling heat generation in solid materials, extending previous one-dimensional results.

Contribution

It establishes the first multi-dimensional global existence result for this thermoviscoelastic system without smallness restrictions on initial data.

Findings

Global weak solutions exist for large initial data in multi-dimensional domains.

The results extend previous one-dimensional findings to higher dimensions.

No smallness condition on initial data is required for existence.

Abstract

This paper investigates a quasilinear parabolic system arising in thermoviscoelasticity of Kelvin-Voigt type with temperature-dependent viscosity and coupled terms. The system, given by \begin{equation*} \begin{cases} u_{tt}=\nabla\cdot\big(\gamma(\Theta)\nabla u_t\big)+a\Delta u-\nabla\cdot f(\Theta), & x \in \Omega,\ t > 0, \Theta_t=\Delta\Theta+\gamma(\Theta)|\nabla u_t|^2-f(\Theta)\nabla u_t, & x \in \Omega,\ t > 0, u=0,\quad\frac{\partial\Theta}{\partial\nu}=0, & x \in \partial\Omega,\ t > 0, u(x,0)=u_0(x),\; u_t(x,0)=u_{0t}(x),\;\Theta(x,0)=\Theta_0(x), & x \in \Omega, \end{cases} \end{equation*} models heat generation by acoustic waves in solid materials and can be derived as a scalar simplification of more complex piezoelectric-thermoviscoelastic model. Under the assumptions that $u_0\in H_0^1(\Omega)$, $u_{0t}\in L^2(\Omega)$, $\Theta_0\in L^1(\Omega)$ with $\Theta_0\geqslant0$ a.e.~in $\Omega$, that $\gamma,f\in C^0([0,\infty))$ satisfy $f(0)=0$, and that there exist constants $k_\gamma,K_\gamma,K_f>0$ and $0<\alpha<\frac{N+2}{2N}$ such that $$k_\gamma\leqslant\gamma(\xi)\leqslant K_\gamma\quad\text{and}\quad |f(\xi)|\leqslant K_f(1+\xi)^\alpha\qquad\forall~\xi\geqslant0,$$ we establish the global existence of weak solutions for arbitrarily large initial data in bounded domains $\Omega\subset\mathbb{R}^N$ ($N\geqslant1$). The result extends recent one-dimensional finding \cite{WinklerZAMP} to the multi-dimensional setting without requiring any smallness condition on the data.