Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by $W^{1,p}$ energy analysis
Leander Claes, Michael Winkler
TL;DR
The paper analyzes a coupled $u_{tt}$–$\Theta_t$ system in a bounded domain with temperature-dependent coefficients $\gamma(\Theta)$ and $\Gamma(\Theta)$. It first establishes local solvability through a Schauder-fixed point framework applied to the reformulated system with $v=u_t+au$, together with a natural extensibility criterion. For $p>n$, it then shows that small-data trajectories yield a global-in-time, classical solution with exponential decay of the gradients $\nabla v$, $\nabla u$, and $\nabla \Theta$ in $L^p$, by deriving a gradient-energy ODI in which linear dissipation dominates the nonlinear heat-production terms. The analysis relies on a Poincaré-type inequality for $| abla\varphi|^p$ and works in any dimension, provided the data are sufficiently small and $\Omega$ is convex. Overall, the work provides a smooth small-data theory for a temperature-dependent quasilinear hyperbolic-parabolic system, highlighting a dissipative structure beyond constant-coefficient models.
Abstract
In bounded $n$-dimensonal domains with $n\ge 1$, this manuscript examines an initial-boundary value problem for the system \[ \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (γ(Θ) \nabla u_t) + a \nabla \cdot (γ(Θ) \nabla u) + \nabla\cdot f(Θ), Θ_t = DΔΘ+ Γ(Θ) |\nabla u_t|^2 + F(Θ)\cdot \nabla u_t, \end{array} \right. \] which in the case $n=1$ and with $γ\equiv Γ$ as well as $f\equiv F$ reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides $f$ and $F$, also the core ingredients $γ$ and $Γ$ may depend on the temperature variable $Θ$. Firstly, a statement on local existence of classical solutions is derived for arbitrary $a>0, D>0$ as well as $0<γ\in C^2([0,\infty))$ and $0\leΓ\in C^1([0,\infty))$, for functions $f\in C^2([0,\infty);{\mathbb{R}}^n)$ and $F\in C^1([0,\infty);{\mathbb{R}}^n)$ with $F(0)=0$, and for suitably regular initial data of arbitrary size. Secondly, it is seen that for each $p\ge 2$ such that $p>n$ there exists $δ(p)>0$ with the property that whenever in addition to the above we have \[ \frac{a}{γ(0)} \le δ(p) \qquad \mbox{and} \qquad \frac{|f'(Θ_\star)| \cdot |F(Θ_\star)|}{D \cdot γ(Θ_\star)} \le δ(p), \] for initial data suitably close to the constant level given by $u=0$ and $Θ=Θ_\star$, with any fixed $Θ_\star\ge 0$, these solutions are actually global in time and have the property that $\nabla u_t, \nabla u$ and $\nablaΘ$ decay exponentially fast in $L^p$. This is achieved by detecting suitable dissipative properties of functionals involving norms of these gradients in $L^p$ spaces.