Damped wave equation with super critical nonlinearities

By Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Differential and Integral Equations, 2004

90 citations recorded by the source corpus.

We study global existence in time of small solutions to the Cauchy problem for the nonlinear damped wave equation ∂2 t u+ ∂tu−∆u = N (u) , x ∈ Rn, t > 0, u(0, x) = eu0 (x) , ∂tu(0, x) = eu1 (x) , x ∈ Rn, (1) where e > 0. The nonlinearity N (u) ∈ Ck (R) satisfies the estimate d j duj N (u) ≤ C |u|ρ−j , 0 ≤ j ≤ k ≤ ρ. The power ρ > 1+ 2 n is considered as super critical for large time. We assume that the initial data u0 ∈ H ∩ H , u1 ∈ Hα−1,0 ∩ H , where δ > n 2 , [α] ≤ ρ, α ≥ n 2 − 1 ρ−1 for n ≥ 2 and α ≥ 12 − 1 2(ρ−1) for n = 1. Weighted Sobolev spaces are H = n φ ∈ L; 〈x〉 〈i∂x〉 φ (x) L2 0 such that for any e ∈ (0, e0] there exists a unique global solution u ∈ C [0,∞) ;Hα,0 ∩H0,δ for the Cauchy problem (1) and solutions satisfy the time decay property ‖u (t)‖Lp ≤ Ct − 2 1− 1 p for all t > 0, where 2 ≤ p ≤ 2n n−2α if α n 2 .

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/16a6462e8b62001e1ed7817f0719e67107659290

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Huijiang Zhao
  2. Masahiro Ikeda
  3. Kenji Nishihara
  4. Takahisa Inui
  5. Yuta Wakasugi
  6. Mamoru Okamoto
  7. Ryo Ikehata
  8. M. DʼAbbicco

In Control and Systems Engineering

  1. Huijiang Zhao
  2. Masahiro Ikeda
  3. Kenji Nishihara
  4. Yuta Wakasugi
  5. Takahisa Inui
  6. Mamoru Okamoto
  7. Ryo Ikehata
  8. M. DʼAbbicco

In Mathematical Physics

  1. Huijiang Zhao
  2. Masahiro Ikeda
  3. Kenji Nishihara
  4. Yuta Wakasugi
  5. Takahisa Inui
  6. Mamoru Okamoto
  7. Ryo Ikehata
  8. M. DʼAbbicco

Citations in the corpus