Acta mathematica scientia, Series B >
DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN#br# MULTI-DIMENSIONAL HALF SPACE
Received date: 2009-07-27
Online published: 2011-07-20
Supported by
The research of Fan Lili was supported by two grants from the National Natural Science Foundation of China (10871151; 10925103), the research of Liu Hongxia was supported by National Natural Science Foundation of China (10871082), and the research of Yin Hui was supported by National Natural Sciences Foundation of China (10901064).
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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utt − Δu + ut + divf(u) = 0, t > 0, x = (x1, x′)∈ Rn+(:= R+ × Rn−1),
u(0, x) = u0(x) → u+, as x1 →+∞,
ut(0, x) = u1(x), u(t, 0, x′) = ub, x′ = (x2, x3, · · · , xn) ∈ 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(ub) < 0. The main purpose of this paper devoted to discussing the case of f′1(ub) ≥ 0 and we show that similar results still hold for
such a case. Our analysis is based on some delicate energy estimates.
FAN Li-Li , LIU Hong-Xia , YIN Hui . DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN#br# MULTI-DIMENSIONAL HALF SPACE[J]. Acta mathematica scientia, Series B, 2011 , 31(4) : 1389 -1410 . DOI: 10.1016/S0252-9602(11)60326-3
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