Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting

TL;DR

This paper proves the global existence and stability of solutions for a quasilinear model of viscous acoustic wave propagation with temperature-dependent parameters, under certain conditions on the model functions and parameters.

Contribution

It establishes conditions ensuring global classical solutions and their stabilization, extending previous results by identifying specific criteria for global existence and long-term behavior.

Findings

Global classical solutions exist under specified conditions.

Solutions stabilize to equilibrium over time.

Finite-time blow-up can occur if conditions are not met.

Abstract

The manuscript considers the model for conversion of mechanical energy into heat during acoustic wave propagation in the presence of temperature-dependent elastic parameters, as given by \[ \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x, \\[1mm] \Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2. \end{array} \right. \qquad \qquad (\star) \] It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $\Omega$, under the assumption that $\gamma\in C^2([0,\infty))$ is such that $\gamma>0$ and $\gamma'\ge 0$ on $[0,\infty)$ as well as \[ D\cdot (\gamma+D) \cdot \gamma'' + 2\gamma \gamma'^2 \le 0 \qquad \mbox{on } [0,\infty), \] for all suitably regular initial data this problem admits a globally defined classical solution. This complements recent findings in the literature, according to which ($\star$) may admit solutions blowing up in finite time whenever $\gamma$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{d\xi}{\gamma(\xi)} < \infty$. Apart from that, it is found that if the additional assumption \[ a|\Omega|^2 \le \frac{\pi^2 \gamma(0)}{1+\sqrt{1+\frac{\gamma(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit.