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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:250

Article

Ueda, Yoshihiro¹  Nakamura, Tohru²  Kawashima, Shuichi² 

[1] Tohoku Univ, Math Inst, Sendai, Miyagi 9808578, Japan

[2] Kyushu Univ, Fac Math, Fukuoka 8190395, Japan

关键词: Energy method;    Fourier transform;    Asymptotic stability;    Planar stationary wave;    Damped wave equation;   

DOI  :  10.1016/j.jde.2010.10.003

来源: Elsevier

【 摘 要 】

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+ x 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' is an element of Rn-1. For the variable x(i) is an element of 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 -> infinity. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13]. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free