Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering $W^{1,q}$-small initial data

In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-