Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by $W^{1,p}$ energy analysis

Abstract

In bounded $n$-dimensonal domains with $n\ge 1$, this manuscript examines an initial-boundary value problem for the system \[ \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma(\Theta) \nabla u_t) + a \nabla \cdot (\gamma(\Theta) \nabla u) + \nabla\cdot f(\Theta), \Theta_t = D\Delta\Theta + \Gamma(\Theta) |\nabla u_t|^2 + F(\Theta)\cdot \nabla u_t, \end{array} \right. \] which in the case $n=1$ and with $\gamma\equiv \Gamma$ 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 $\gamma$ and $\Gamma$ may depend on the temperature variable $\Theta$. Firstly, a statement on local existence of classical solutions is derived for arbitrary $a>0, D>0$ as well as $0<\gamma\in C^2([0,\infty))$ and $0\le\Gamma\in C^1([0,\infty))$, 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 $\delta(p)>0$ with the property that whenever in addition to the above we have \[ \frac{a}{\gamma(0)} \le \delta(p) \qquad \mbox{and} \qquad \frac{|f'(\Theta_\star)| \cdot |F(\Theta_\star)|}{D \cdot \gamma(\Theta_\star)} \le \delta(p), \] for initial data suitably close to the constant level given by $u=0$ and $\Theta=\Theta_\star$, with any fixed $\Theta_\star\ge 0$, these solutions are actually global in time and have the property that $\nabla u_t, \nabla u$ and $\nabla\Theta$ 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.