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

TL;DR

This paper establishes the global existence, uniqueness, and exponential stability of solutions to Kuznetsov's equation in L_p-spaces, with optimal regularity conditions and simplified proofs, advancing nonlinear acoustics modeling.

Contribution

It provides the first proof of global well-posedness and exponential stability for Kuznetsov's equation under optimal regularity assumptions using maximal L_p-regularity and the implicit function theorem.

Findings

Existence of a unique global solution for small initial data.

Solutions and their derivatives decay exponentially over time.

Simplified proof techniques based on maximal L_p-regularity.

Abstract

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.