Almost Global Existence for 3-D Quasilinear Wave Equations in Exterior Domains with Neumann Boundary Conditions

Abstract

<p style='text-indent:20px;'>We are concerned with the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. When the surfaces of 3-D compact convex obstacles are smooth and the quadratic nonlinearities in the quasilinear wave equations do not fulfill the null condition, we establish the almost global existence of smooth small-amplitude solutions to the above initial-boundary value problems. The lower bound of the lifespan is proved to be <inline-formula><tex-math id="M1">\begin{document}$ e^{\kappa/\varepsilon} $\end{document}</tex-math></inline-formula> with some positive <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> and the small positive parameter <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> as the size of the initial data. This lower bound is optimal as shown in the 3-D boundaryless case.</p>