Damped wave equation with a critical nonlinearity
Damped wave equation with a critical nonlinearity
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DOI：
10.1090/s0002-9947-05-03818-3
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发表时间：
2005-04
影响因子：
1.3
通讯作者：
N. Hayashi;E. Kaikina;P. Naumkin
中科院分区：
文献类型：
--
作者：
N. Hayashi;E. Kaikina;P. Naumkin
研究了具有临界非线性项的非线性阻尼波方程Cauchy问题小解的大时间渐近性{<$2tu +<$tu- Δu + λ u1 + 2n = 0，x ∈ Rn，t > 0，u（0，x）= eu 0（x），<$tu（0，x）= eu 1（x），x ∈ Rn，其中e > 0，空间维数n = 1，2,3.设初值uo ∈ H δ，0 nH 0，δ，u1 eH δ-1，0 nH-1，δ，其中δ > n2，加权Sobolev空间为H1，m = {Φ eL 2; m1 Φ（x）<$L2 =<$1 + x2.并假设λθ 2/n > 0，<$u0（x）dx > 0，其中证明了存在一个正的e0使得上述Cauchy问题有唯一的整体解u ∈ C（[0，oo）; H δ，0）满足时间衰减性质<$u（t）-eθG（t，x）e -φ（t）<$Lp ≤ Ce 1+ 2ng-1-n2（t）-n2（1-1/p），其中e ∈（0，e0].
We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂ 2 t u + ∂ t u - Δu + λu 1+2 n = 0, x ∈ R n , t > 0, u(0, x) = eu 0 (x), ∂ t u(0, x) = eu 1 (x), x ∈ R n , where e > 0, and space dimensions n = 1, 2,3. Assume that the initial data uo ∈ H δ,0 n H 0,δ , u 1 e H δ-1,0 n H -1,δ , where δ > n 2, weighted Sobolev spaces are H l,m = {Φ e L2; m l Φ(x)∥ L 2 = √1 + x 2 . Also we suppose that λθ 2/n > 0, ∫u 0 (x) dx > 0, where Then we prove that there exists a positive e 0 such that the Cauchy problem above has a unique global solution u ∈ C ([0, oo); H δ,0 ) satisfying the time decay property ∥u(t)-eθG(t,x)e -φ(t) ∥ Lp ≤ Ce 1+2 n g -1-n 2 (t) -n 2(1-1/p) for all t > 0, 1 ≤ p ≤ ∞, where e ∈ (0, e 0 ].