Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation
Rainer Brunnhuber Barbara Kaltenbacher
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4515-4535 doi: 10.3934/dcds.2014.34.4515
We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.
    We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.
    Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.
    Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.
keywords: asymptotic behavior. well-posedness Nonlinear acoustics
Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling
Rainer Brunnhuber Barbara Kaltenbacher Petronela Radu
Evolution Equations & Control Theory 2014, 3(4): 595-626 doi: 10.3934/eect.2014.3.595
In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
keywords: nonlinear damping quasilinear wave equation local existence. Nonlinear acoustics

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