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

Abstract

In a smoothly bounded domain $\Om\subset\R^n$ with $n\ge 1$ 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-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_0,u_{0t})=(u,u_t){|}_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_t)$ in $W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.