The anti-derivative method in the half space and application to damped wave equations with non-convex convection

摘要

We study the asymptotic stability of nonlinear waves for damped wave equations with a non-convex convection term on the half line. In the case where the convection term satisfies the convexity and sub-characteristic conditions at the origin, it is shown by our previous work [Osaka J. Math. To appear] that the solution tends to the stationary wave. In this paper, we deal with the damped wave equations with convection term which is not necessarily the convexity at the origin, and we show not only the asymptotic stability of the non-degenerate stationary wave but also for the degenerate stationary wave. Furthermore, for the non-degenerate case, we also show that the time convergence rate is polynomially (respectively exponentially) fast if the initial perturbation decays polynomially (respectively exponentially) as x goes up to infinity. Our proofs are based on a combination of the antiderivative method and the L~2 weighted energy method.