## Abstract

Consider the Cauchy problem for nonlinear dissipative evolution equations {(u_{t} + N (u, u) + L u = 0,, x ∈ R, t > 0,; u (0, x) = u_{0} (x),, x ∈ R,) where L is the linear pseudodifferential operator L u = over(F, -)_{ξ → x} (L (ξ) over(u, ̂) (ξ)) and the nonlinearity is a quadratic pseudodifferential operator N (u, u) = over(F, -)_{ξ → x} ∫_{R} A (t, ξ, y) over(u, ̂) (t, ξ - y) over(u, ̂) (t, y) d y,over(u, ̂) ≡ F_{x → ξ} u is direct Fourier transformation. Let the initial data u_{0} ∈ H^{β, 0} ∩ H^{0, β}, β > frac(1, 2), are sufficiently small and have a non-zero total mass M = ∫ u_{0} (x) d x ≠ 0, here H^{n, m} = {φ{symbol} ∈ L^{2} {norm of matrix} 〈 x 〉^{m} 〈 i ∂_{x} 〉^{n} φ{symbol} (x) {norm of matrix}_{L2} < ∞} is the weighted Sobolev space. Then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self-similar solution defined uniquely by the total mass M of the initial data.

Original language | English |
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Pages (from-to) | 2914-2933 |

Number of pages | 20 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 67 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2007 Nov 15 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics