Global Smooth Solutions to a Thermoelastic Cauchy Problem in Phase Transitions

TL;DR

This paper proves the global existence and uniqueness of smooth solutions for a thermoelastic phase transition model combining viscoelasticity, heat transfer, and phase change effects.

Contribution

It extends the isothermal phase transition model by coupling it with a heat equation and establishes global well-posedness results for the resulting thermoelastic system.

Findings

Global existence and uniqueness of classical solutions are proven.

Temperature perturbations decay algebraically under certain conditions.

The analysis employs traveling-wave decomposition and energy estimates.

Abstract

We study one-dimensional viscoelastic phase transitions modeled by a Ginzburg--Landau energy with a non-convex cubic stress-strain law. Extending the isothermal model, we couple the momentum equation to a heat equation for the temperature field, giving a thermoelastic system with viscous, capillary, and thermal-diffusion terms. We prove global existence and uniqueness of classical smooth solutions for the Cauchy problem, using a traveling-wave decomposition, an exponential transformation of the mechanical perturbation, and coupled energy estimates at successive regularity levels. Under additional integrability and small-data assumptions, the temperature perturbation decays algebraically.