On wellposedness in nonlinear acoustics

Motivated by a medical application from lithotripsy, we study the initial–boundary value problem given by Westervelt equation (1) in a bounded domain Ω. This models the nonlinear evolution of the acoustic pressure u excited at a part Γ₀ of the boundary. Along with the excitation given by Neumann boundary condition as in (1) , we also consider the Dirichlet type of excitation. Whereas shock waves are known to emerge after a sufficiently large time interval for appropriate initial and boundary conditions, we here prove existence and uniqueness as well as stability of a solution u for small data g, u₀ and u₁ or short time T, using a fixed point argument. Moreover we extend the result to the more general model given by the Kuznetsov equation (2) for the acoustic velocity potential ψ. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)