Abstract
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].
| Original language | English |
|---|---|
| Pages (from-to) | 1169-1199 |
| Number of pages | 31 |
| Journal | Journal of Differential Equations |
| Volume | 250 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - Jan 15 2011 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
Cite this
Energy method in the partial Fourier space and application to stability problems in the half space. / Ueda, Yoshihiro; Nakamura, Tohru; Kawashima, Shuichi.
In: Journal of Differential Equations, Vol. 250, No. 2, 15.01.2011, p. 1169-1199.Research output: Contribution to journal › Article
}
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
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].
UR - http://www.scopus.com/inward/record.url?scp=78149470321&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=78149470321&partnerID=8YFLogxK
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 -