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

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(γ(Θ)\nabla u_t\big)+aΔu-\nabla\cdot f(Θ), & x \in Ω,\ t > 0, Θ_t=ΔΘ+γ(Θ)|\nabla u_t|^2-f(Θ)\nabla u_t, & x \in Ω,\ t > 0, u=0,\quad\frac{\partialΘ}{\partialν}=0, & x \in \partialΩ,\ t > 0, u(x,0)=u_0(x),\; u_t(x,0)=u_{0t}(x),\;Θ(x,0)=Θ_0(x), & x \in Ω, \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 $u_0\in H_0^1(\hat{I}\text{c◯})$, $u_{0t}\in L^2(\hat{I}\text{c◯})$, $\hat{I}{˜}_0\in L^1(\hat{I}\text{c◯})$ with $\hat{I}{˜}_0⩾0$ a.e.~in $\hat{I}\text{c◯}$, that ${\hat{I}}^3,f\in C^0([0,\infty ))$ satisfy $f(0)=0$, and that there exist constants $k_{\hat{I}}^3,K_{\hat{I}}^3,K_f>0$ and $0<\hat{I}\pm <\frac{N+2}{2N}$ such that $$k_{\hat{I}}^3⩽{\hat{I}}^3(\hat{I}¾)⩽K_{\hat{I}}^3\text{and}|f(\hat{I}¾)|⩽K_f(1+\hat{I}¾{)}^{\hat{I}}\pm \forall \hat{I}¾⩾0,$$ we establish the global existence of weak solutions for arbitrarily large initial data in bounded domains $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ ($N⩾1$). The result extends recent one-dimensional finding \cite{WinklerZAMP} to the multi-dimensional setting without requiring any smallness condition on the data.