Almost Global Existence for Quasilinear Wave Equations in Waveguides with Neumann Boundary Conditions
نویسنده
چکیده
Abstract. In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the Laplacian of the compact base in terms of its eigenfunctions. For those terms corresponding to zero modes, we obtain decay using analogs of estimates of Klainerman and Sideris. For the nonzero modes, estimates for Klein-Gordon equations, which provide better decay, are available.
منابع مشابه
ar X iv : m at h / 04 11 51 3 v 2 [ m at h . A P ] 1 4 A pr 2 00 5 NONLINEAR HYPERBOLIC EQUATIONS IN INFINITE HOMOGENEOUS WAVEGUIDES
In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neu-mann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results tha...
متن کاملar X iv : m at h / 04 11 51 3 v 1 [ m at h . A P ] 2 3 N ov 2 00 4 NONLINEAR HYPERBOLIC EQUATIONS IN INFINITE HOMOGENEOUS WAVEGUIDES
In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neu-mann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results tha...
متن کاملWell-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions
In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations wit...
متن کاملAlmost Global Existence for Quasilinear Wave Equations in Three Space Dimensions
This article studies almost global existence for solutions of quadratically quasilinear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichletwave equations in the exterior of a smooth, sta...
متن کاملGlobal existence, stability results and compact invariant sets for a quasilinear nonlocal wave equation on $mathbb{R}^{N}$
We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type [ u_{tt}-phi (x)||nabla u(t)||^{2}Delta u+delta u_{t}=|u|^{a}u,, x in mathbb{R}^{N} ,,tgeq 0;,]with initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N geq 3, ; delta geq 0$ and $(phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(mathb...
متن کامل