关键词: Damped wave equation, planar stationary wave, a priori estimates, decay rates, space-time weighted energy method

Abstract:

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space Rn
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ut_t − Δu + u_t + divf(u) = 0, t > 0, x = (x₁, x′)∈ Rⁿ₊(:= R₊ × Rⁿ⁻¹),
u(0, x) = u₀(x) → u₊, as x₁ →+∞,
u_t(0, _x) = u₁(x), u(t, 0, x′) = u_b, x′ = (x₂, x₃, · · · , x_n) ∈ Rⁿ−1. (I)
For the non-degenerate case f′₁(u₊) < 0, it was shown in [10] that the above initial-boundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave Φ(x₁) uniformly in x₁ ∈R₊ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t →+∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f′₁(u_b) < 0. The main purpose of this paper devoted to discussing the case of f′₁(u_b) ≥ 0 and we show that similar results still hold for
such a case. Our analysis is based on some delicate energy estimates.

Key words: Damped wave equation, planar stationary wave, a priori estimates, decay rates, space-time weighted energy method