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

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 \[ \left\{ \begin{array}{l} u_{tt} = (γ(Θ) u_{xt})_x + a (γ(Θ) u_x)_x, \\[1mm] Θ_t = DΘ_{xx} + γ(Θ) u_{xt}^2. \end{array} \right. \qquad \qquad (\star) \] 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 ${\gamma }^{\prime }\ge 0$ on $[0,\infty )$ as well as \[ D\cdot (γ+D) \cdot γ'' + 2γγ'^2 \le 0 \qquad \mbox{on } [0,\infty), \] 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 \[ a|Ω|^2 \le \frac{π^2 γ(0)}{1+\sqrt{1+\frac{γ(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit.