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Hotspot formation driven by temperature-dependent coefficients in one-dimensional thermoviscoelasticity

Michael Winkler

TL;DR

The paper analyzes a one-dimensional thermoviscoelastic model with temperature-dependent viscosity and stiffness, introducing $\gamma(\Theta)$ and $a\gamma(\Theta)$. It develops a local solvability framework and a refined extensibility criterion showing that finite-time blow-up in the solution obliges $\Theta$ to become unbounded, i.e., hotspot formation. By transforming to $v=u_t+au$ and employing a Moser-type iteration, it derives $L^{\infty}$ bounds and gradient controls that underpin the blow-up analysis. The main results establish finite-time blow-up under superlinear growth of $\gamma$ through two mechanisms: large initial energy and large initial temperature, providing precise thresholds and highlighting hotspot formation as a fundamental feature of temperature-dependent thermoviscoelastic systems with Kelvin-Voigt-type damping.

Abstract

This manuscript is concerned with a two-component evolution system generalizing the classical model for one-dimensional thermoviscoelastic dynamics in Kelvin-Voigt materials in the presence of temperature-dependent viscosities and elastic stiffnesses. Under suitable assumptions on the growth of these ingredients and on the initial data, the occurrence of finite-time blow-up with respect to the $L^\infty$ norm in the temperature variable is discovered.

Hotspot formation driven by temperature-dependent coefficients in one-dimensional thermoviscoelasticity

TL;DR

The paper analyzes a one-dimensional thermoviscoelastic model with temperature-dependent viscosity and stiffness, introducing and . It develops a local solvability framework and a refined extensibility criterion showing that finite-time blow-up in the solution obliges to become unbounded, i.e., hotspot formation. By transforming to and employing a Moser-type iteration, it derives bounds and gradient controls that underpin the blow-up analysis. The main results establish finite-time blow-up under superlinear growth of through two mechanisms: large initial energy and large initial temperature, providing precise thresholds and highlighting hotspot formation as a fundamental feature of temperature-dependent thermoviscoelastic systems with Kelvin-Voigt-type damping.

Abstract

This manuscript is concerned with a two-component evolution system generalizing the classical model for one-dimensional thermoviscoelastic dynamics in Kelvin-Voigt materials in the presence of temperature-dependent viscosities and elastic stiffnesses. Under suitable assumptions on the growth of these ingredients and on the initial data, the occurrence of finite-time blow-up with respect to the norm in the temperature variable is discovered.
Paper Structure (7 sections, 19 theorems, 142 equations)

This paper contains 7 sections, 19 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. The role of $\Theta$ near possible singularities. Proof of Proposition \ref{['prop_ext']}
  3. Local-in-time $L^\infty$ bounds for $u$ and $u_t$
  4. First-oder testing procedures. A refined extensibility criterion
  5. Blow-up in the presence of superlinear $\gamma$
  6. A necessary condition on $(u_0,u_{0t},\Theta_0)$ for existence up to $t=T$
  7. Conclusion. Proofs of Theorem \ref{['theo65']} and Theorem \ref{['theo66']}

Key Result

Proposition 1.1

Let $n\ge 1$ and $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, let $a>0$ and $D>0$, and suppose that Then whenever one can find $T_{max}\in (0,\infty]$ as well as such that $\Theta\ge 0$ in $\Omega\times (0,T_{max})$, that $(u,\Theta)$ forms a classical solution of (0) in $\Omega\times (0,T_{max})$, and that

Theorems & Definitions (19)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 9 more