| 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 2 - The nonlinear problem can be solved with the implicit function theorem, since its linearization has maximal regularity and therefore generates a topological isomorphism between the space of solutions and the space of data. |
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