DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

摘要

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+:｛ utt-Δu+ut+divf(u)=0,t＞0,x=(x1,x′)∈R+(:=R+ ×R(n-1)),u(0,x) =u0(x) → u+,as x1 → +∞,(Ⅰ)x′ ut(0,x) =u1(x),u(t,0,x′) =ub,=(x2,x3,…,xn) ∈ Rn-1.For the non-degenerate case f′1(u+) ＜ 0,it was shown in[10]that the above initialboundary 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 → +oo 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 is 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.