Abstract
We study the asymptotic stability of nonlinear waves for damped wave equations with a convection term on the half line. In the case where the convection term satisfies the convex and sub-characteristic conditions, it is known by the work of Ueda [7] and Ueda--Nakamura--Kawashima [10] that the solution tends toward a stationary solution. In this paper, we prove that even for a quite wide class of the convection term, such a linear superposition of the stationary solution and the rarefaction wave is asymptotically stable. Moreover, in the case where the solution tends to the non-degenerate stationary wave, we derive that the time convergence rate is polynomially (resp. exponentially) fast if the initial perturbation decays polynomially (resp. exponentially) as $x \to \infty$. Our proofs are based on a technical $L^{2}$ weighted energy method.
Citation
Itsuko Hashimoto. Yoshihiro Ueda. "Asymptotic behavior of solutions for damped wave equations with non-convex convection term on the half line." Osaka J. Math. 49 (1) 37 - 52, March 2012.
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