TY - JOUR
T1 - Energy method in the partial Fourier space and application to stability problems in the half space
AU - Ueda, Yoshihiro
AU - Nakamura, Tohru
AU - Kawashima, Shuichi
N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2011/1/15
Y1 - 2011/1/15
N2 - The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in R+n=R+×Rn-1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x'∈Rn-1. For the variable x1∈R+ in the normal direction, we use L2 space or weighted L2 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].
AB - The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in R+n=R+×Rn-1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x'∈Rn-1. For the variable x1∈R+ in the normal direction, we use L2 space or weighted L2 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].
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U2 - 10.1016/j.jde.2010.10.003
DO - 10.1016/j.jde.2010.10.003
M3 - Article
AN - SCOPUS:78149470321
VL - 250
SP - 1169
EP - 1199
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 2
ER -