Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid

Abstract

This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} \tau u_{ttt} + \alpha u_{tt} = b \big(\gamma(\Theta) u_{xt}\big)_x + \big( \gamma(\Theta) u_x\big)_x, \\[1mm] \Theta_t = D \Theta_{xx} + b\gamma(\Theta) u_{xt}^2, \end{array} \] 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 $\tau>0, b>0$ and $\alpha\geq0$ satisfy \begin{align}\tag{$\star$} \alpha b >\tau, \end{align} it is shown that for all $\Theta_\star>0$ one can find $\nu=\nu(D,\tau,\alpha,b,\Theta_\star,\gamma)>0$ 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 $$\|\Theta_0\|_{L^\infty(\Omega)}\leq \Theta_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_\Omega u_{0xx}^2 + \int_\Omega (u_{0t})_{xx}^2 + \int_\Omega (u_{0tt})_x^2 < \nu\quad\text{and}\quad \|\Theta_{0x}\|_{L^\infty(\Omega)} + \|\Theta_{0xx}\|_{L^\infty(\Omega)} < \nu.$$ 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.