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, Θt=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)=Θ0(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 $u_0\in H_0^1(Ω)$, $u{0t}\in L^2(Ω)$, $Θ0\in L^1(Ω)$ with $Θ_0\geqslant0$ a.e.~in $Ω$, that $γ,f\in C^0([0,\infty))$ satisfy $f(0)=0$, and that there exist constants $kγ,K_γ,K_f>0$ and $0<α<\frac{N+2}{2N}$ such that $$k_γ\leqslantγ(ξ)\leqslant K_γ\quad\text{and}\quad |f(ξ)|\leqslant K_f(1+ξ)^α\qquad\forall~ξ\geqslant0,$$ we establish the global existence of weak solutions for arbitrarily large initial data in bounded domains $Ω\subset\mathbb{R}^N$ ($N\geqslant1$). The result extends recent one-dimensional finding \cite{WinklerZAMP} to the multi-dimensional setting without requiring any smallness condition on the data.