Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid
Tobias Black, Michael Winkler
TL;DR
The paper studies a one-dimensional thermoacoustic system featuring a Moore–Gibson–Thompson-type third-order-in-time displacement equation coupled to a temperature field through temperature-dependent coefficients. By introducing diffusion-regularized approximations and constructing Lyapunov-type energy functionals, the authors establish global existence and exponential stabilization under the dissipation-dominated regime $αb>τ$, with robust bounds that do not require smallness of the initial temperature in $L^\infty$. The analysis proceeds via a two-phase approach: (i) derive uniform-in-$ε$ a priori estimates and a conditional energy property for the mechanical part, and (ii) pass to the limit $ε o0$ using Aubin–Lions to obtain a global strong solution of the original coupled system. The results show that the temperature becomes spatially homogeneous up to an exponentially decaying remainder and converges to a positive steady state $Θ_\infty$, highlighting strong dissipation despite nonlinear coupling to temperature. This extends the global stability understanding of MGT-type models to nonlinear thermoelastic systems with temperature-dependent parameters.
Abstract
This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} τu_{ttt} + αu_{tt} = b \big(γ(Θ) u_{xt}\big)_x + \big( γ(Θ) u_x\big)_x, \\[1mm] Θ_t = D Θ_{xx} + bγ(Θ) u_{xt}^2, \end{array} \] which is connected to the simplified modeling of heat generation in Zener type materials subject to stress from acoustic waves. Under the assumption that the coefficients $τ>0, b>0$ and $α\geq0$ satisfy \begin{align}\tag{$\star$} αb >τ, \end{align} it is shown that for all $Θ_\star>0$ one can find $ν=ν(D,τ,α,b,Θ_\star,γ)>0$ such that an associated Neumann type initial-boundary value problem with Neumann data admits a unique time-global solution in a suitable framework of strong solvability whenever the initial temperature distribution fulfills $$\|Θ_0\|_{L^\infty(Ω)}\leq Θ_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_Ωu_{0xx}^2 + \int_Ω(u_{0t})_{xx}^2 + \int_Ω(u_{0tt})_x^2 < ν\quad\text{and}\quad \|Θ_{0x}\|_{L^\infty(Ω)} + \|Θ_{0xx}\|_{L^\infty(Ω)} < ν.$$ The constructed solution moreover features an exponential stabilization property for both components. In particular, the parameter range described by ($\star$) coincides with the full stability regime known for the corresponding Moore--Gibson--Thompson equation despite the fairly strong nonlinear coupling to the temperature variable.