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 . In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable . For the variable in the normal direction, we use space or weighted 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 . The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) . [Copyright &y& Elsevier]