Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space

Yoshihiro Ueda; Tohru Nakamura; Shuichi Kawashima

doi:10.3934/krm.2008.1.49

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2008, Volume 1, Issue 1: 49-64. Doi: 10.3934/krm.2008.1.49

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Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space

1.
Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581
2.
Faculty of Mathematics, Kyushu University, Fukuoka 812-8581

Received: October 31, 2007

Revised: October 31, 2007

Published: February 2008

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Abstract

In this paper, we consider the large-time behavior of solutions to the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space. We show that the solution to the problem converges to the corresponding planar stationary wave as time tends to infinity under smallness condition on the initial perturbation. It is proved that the tangential derivatives of the solution verify quantitative decay estimates for $t\to\infty$. Moreover, an additional algebraic convergence rate is obtained by assuming that the initial perturbation decays algebraically in the normal direction. The crucial point of the proof is to derive a priori estimates of solutions by using the time and space weighted energy method.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 35B35, 35B40.

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Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space

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