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Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

Leander Claes, Michael Winkler

TL;DR

This work analyzes a one-dimensional quasilinear Moore-Gibson-Thompson type model for heat generation in thermoviscoelastic media, coupling a third-order-in-time displacement equation with a diffusion-type temperature equation. The authors establish local-in-time existence and uniqueness of strong solutions under Neumann boundary conditions, without small-data assumptions, and provide an extensibility criterion. They employ ε-regularization of temperature-dependent coefficients, energy functionals, Steklov averaging, and Aubin-Lions compactness to pass to the limit and obtain a strong solution, with a blow-up criterion controlling the maximal time of existence. The results contribute a rigorous local well-posedness theory for quasilinear thermo-viscoelastic systems in 1D and lay a foundation for potential extensions to higher dimensions with additional regularity requirements.

Abstract

This manuscript is concerned with the evolution system \[ \left\{ \begin{array}{l} u_{ttt} + αu_{tt} = \big(γ(Θ) u_{xt}\big)_x + \big( \widehatγ(Θ) u_x\big)_x, Θ_t = D Θ_{xx} + Γ(Θ) u_{xt}^2, \end{array} \right. \] which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type. Under the assumptions that $D>0$ and $α\ge 0$, and that $γ, \widehatγ$ and $Γ$ are sufficiently smooth with $γ>0, \widehatγ>0$ and $Γ\ge 0$ on $[0,\infty)$, for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.

Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

TL;DR

This work analyzes a one-dimensional quasilinear Moore-Gibson-Thompson type model for heat generation in thermoviscoelastic media, coupling a third-order-in-time displacement equation with a diffusion-type temperature equation. The authors establish local-in-time existence and uniqueness of strong solutions under Neumann boundary conditions, without small-data assumptions, and provide an extensibility criterion. They employ ε-regularization of temperature-dependent coefficients, energy functionals, Steklov averaging, and Aubin-Lions compactness to pass to the limit and obtain a strong solution, with a blow-up criterion controlling the maximal time of existence. The results contribute a rigorous local well-posedness theory for quasilinear thermo-viscoelastic systems in 1D and lay a foundation for potential extensions to higher dimensions with additional regularity requirements.

Abstract

This manuscript is concerned with the evolution system which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type. Under the assumptions that and , and that and are sufficiently smooth with and on , for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.
Paper Structure (6 sections, 16 theorems, 226 equations)

This paper contains 6 sections, 16 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Strong solutions. Uniqueness
  3. Existence and extensibility for initial data with $\int_\Omega u_0 = \int_\Omega u_{0t}= \int_\Omega u_{0tt}=0$
  4. Approximate problems and their local solvability
  5. Implications of presupposed $L^\infty$ bounds for $\Theta_\varepsilon$ and $\Theta_{\varepsilon t}$
  6. Conclusion

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}$ be an open bounded interval, let $D>0$ and $\alpha\ge 0$, and suppose that Then whenever there exists $T_{max}\in (0,\infty]$ as well as a unique pair $(u,\Theta)$ of functions which are such that and that $\Theta\ge 0$ in $\Omega\times (0,T_{max})$, that $(u,\Theta)$ forms a strong solution of (0) in $\Omega\times (0,T_{max})$ in the sense of Definition dw below

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 7 more