An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay

Abstract

This paper deals with well-posedness of the Kuznetsov equation, which is an enhanced model for nonlinear acoustic wave propagation, e.g., in the context of high intensity ultrasound therapy. This is a quasilinear evolutionary wave equation with potential degeneration and strong damping. We consider it on a bounded domain in R n, n = 1, 2, 3, with possibly inhomogeneous Dirichlet boundary conditions. Based on appropriate energy estimates and Banach's fixed point theorem applied to an appropriate formulation of the PDE, we first of all prove local well-posedness with small initial data. For proving global existence, we use barrier's method and exploit the dissipative mechanism leading to decay rates. The latter also allow us to prove exponential decay. For the treatment of inhomogeneous boundary conditions, appropriate extensions of the boundary data to the interior are crucial. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

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