Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting
Felix Meyer, Michael Winkler
TL;DR
The paper analyzes a 1D quasilinear thermo-viscoelastic model for acoustic wave propagation with temperature-dependent elasticity, represented by $u_{tt} = (\gamma(\Theta) u_{xt})_x + a(\gamma(\Theta) u_x)_x$ and $\Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2$. It proves global existence of classical solutions under the structural condition $D\cdot (\gamma+D) \cdot \gamma'' + 2\gamma \gamma'^2 \le 0$ with $\gamma>0$, $\gamma'\ge 0$, and demonstrates large-time stabilization to a spatially homogeneous state under an additional small-domain bound on $a|\Omega|^2$. The methodology relies on constructing coupled energy functionals $y^{(B)}(t)$ that capture interactions between $u$ and $\Theta$, deriving dissipative inequalities, and employing Grönwall-type arguments to secure global solvability; for large times, a refined energy setup yields exponential decay of $u_x$, $v_x$, and $\Theta$, and convergence of the mean temperature to $\Theta_\infty$. Collectively, the results delineate conditions under which nonlinear heat generation does not induce finite-time blow-up and quantify asymptotic convergence, with implications for material behavior under temperature-dependent parameters.
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 $Ω$, under the assumption that $γ\in C^2([0,\infty))$ is such that $γ>0$ and $γ'\ge 0$ 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 $γ$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{dξ}{γ(ξ)} < \infty$. 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.