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

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 $$\{\begin{array}{l}{u}_{tt}=(\gamma (\mathrm{\Theta })u_{xt}{)}_x+a(\gamma (\mathrm{\Theta })u_x{)}_x,\\ {\mathrm{\Theta }}_t=D{\mathrm{\Theta }}_{xx}+\gamma (\mathrm{\Theta })u_{xt}^2\mathrm{.}\end{array}(\star )$$ It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $\mathrm{\Omega }$, under the assumption that $\gamma \in C^2([0,\mathrm{\infty }))$ is such that $\gamma >0$ and ${\gamma }^{\mathrm{\prime }}\ge 0$ on $[0,\mathrm{\infty })$ as well as $$D\cdot (\gamma +D)\cdot {\gamma }^{\mathrm{\prime }\mathrm{\prime }}+2\gamma {\gamma }^{\mathrm{\prime }2}\le 0on[0,\mathrm{\infty }),$$ 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 $[0,\mathrm{\infty })$ with ${\int }_0^{\mathrm{\infty }}\frac{d\xi }{\gamma (\xi )}<\mathrm{\infty }$. Apart from that, it is found that if the additional assumption $$a\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^2\le \frac{{\pi }^2\gamma (0)}{1+\sqrt{1+\frac{\gamma (0)}{D}}}$$ is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit.