Well-Posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation

Abstract

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.

Cited by in corpus (5)

• Nonlinear Acoustics and Shock Formation in Lossless Barotropic Green--Naghdi Fluids
• Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation
• Asymptotic behaviors for Blackstock's model of thermoviscous flow
• Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions
• Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow