Energy method in the partial Fourier space and application to stability problems in the half space

Abstract

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R n . In this paper, we study half space problems in R + n = R + × R n − 1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x ′ ∈ R n − 1 . For the variable x 1 ∈ R + in the normal direction, we use L 2 space or weighted L 2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ . The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].