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

TL;DR

This paper investigates a generalized one-dimensional thermoviscoelastic model with temperature-dependent coefficients, revealing conditions under which solutions blow up in finite time, thus advancing understanding of material behavior under thermal effects.

Contribution

It introduces a two-component evolution system with temperature-dependent viscosities and elastic stiffnesses, analyzing blow-up phenomena in this generalized setting.

Findings

Finite-time blow-up occurs under certain growth conditions.

Temperature-dependent coefficients significantly influence solution behavior.

The model extends classical thermoviscoelastic equations to more realistic scenarios.

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.