Monday, September 10, 2012

1209.1456 (Stefan Meyer et al.)

Global well-posedness and exponential stability for Kuznetsov's equation
in L_p-spaces

Stefan Meyer, Mathias Wilke
We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal L_p-regularity for parabolic equations and the implicit function theorem.
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