Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities

TL;DR

The paper proves global existence of global weak solutions for a one-dimensional thermoviscoelastic model with temperature-dependent viscosity $\gamma(\Theta)$ and dilation $f(\Theta)$ under the bounds $k_{\gamma} \le \gamma(\xi) \le K_{\gamma}$ and $|f(\xi)| \le K_f (\xi+1)^\alpha$ with $\alpha<\tfrac{3}{2}$ and $f(0)=0$. It employs an energy-consistent parabolic regularization with parameter $\varepsilon$, deriving a dissipative energy identity and uniform $\varepsilon$-independent estimates. Through Aubin-Lions compactness and a delicate limit passage for the quadratic heat-production term using Steklov averages, it constructs a limit triple $(v,u,\Theta)$ solving the original system in a weak sense. The results advance the mathematical understanding of thermo-viscoelastic systems with temperature-dependent coefficients and have potential implications for related piezoelectric-thermoviscoelastic models in 1D.

Abstract

An initial-boundary value problem for \[ \left\{ \begin{array}{ll} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \qquad & x\inΩ, \ t>0, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \qquad & x\inΩ, \ t>0, \end{array} \right. \] is considered in an open bounded real interval $Ω$. Under the assumption that $γ\in C^0([0,\infty))$ and $f\in C^0([0,\infty))$ are such that $f(0)=0$, and $k_γ\le γ\le K_γ$ as well as \[ |f(ξ)| \le K_f \cdot (ξ+1)^α \qquad \mbox{for all } ξ\ge 0 \] with some $k_γ>0, K_γ>0, K_f>0$ and $α<\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4