| Contents |
| We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation
\begin{align*}
u_{tt} - c^2\Delta_x u - b \Delta_x u_t &= k (u^2)_{tt}+\rho_0(v\cdot v)_{tt},
\end{align*}
with a non-homogeneous Dirichlet boundary condition. This model describes the propagation of sound in a fluidic medium when the external pressure is prescribed. We prove that for small initial and boundary data there exists a unique global solution with optimal $L_p$-regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal $L_p$-regularity for quasilinear parabolic equations.
Part 1 - The linearized problem. The initial-boundary value problems for both Westervelt's and Kuznetsov's equations can be reduced to a common linearized problem with vanishing boundary data. We show that this problem is uniquely solvable with maximal regularity and the solutions decay exponentially. Similar assertions are proved for related linear problems which are needed to solve the nonlinear problem for Kuznetsov's equation. |
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