Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering W1,q-small initial data
Abstract
In a smoothly bounded domain Om⊂Rn with n≥ 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 (u0, u0t)=(u, ut)|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,ut) 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.
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BibTeX
@misc{boudjeriou2026global,
title = {{Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering $W^1,q$-small initial data}},
author = {Tahir Boudjeriou and Michael Winkler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02795},
eprint = {2609.02795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02795},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering W^1,q-small initial data
AU - Tahir Boudjeriou
AU - Michael Winkler
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02795
UR - https://arxiv.org/abs/2609.02795
AB - In a smoothly bounded domain Om⊂R^n with n≥ 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.
ER -