## Abstract

We study the Cauchy problem for non-linear dissipative evolution equations ℒ(u_{t} + script N sign(u) + ℒu = 0, x ∈ ℝ, t>0 u(0,x) = u_{0}(x), x ∈ ℝ where ℒ is the linear pseudodifferential operator ℒu = ℱ̄ _{ξ→x}(L(ξ)û(ξ)) and the non-linearity is a quadratic pseudodifferential operator script N sign(u) = ℱ̄ _{ξ→x} ∫_{ℝ} a(t, ξ, y)û(t,ξ - y)û(t,y)dy û ≡ ℱ_{x→ξ}u is the Fourier transformation. We consider non-convective type non-linearity, that is we suppose that a(t,0,y) ≠ 0. Let the initial data u_{0} ∈ H ^{q,0} ∩ H^{0, q}, q> 1/2, are sufficiently small and have a non-zero total mass ∫ u_{0}(x)dx ≠ 0, where H ^{n,m} = {φ ∈ L^{2}∥〈x〉 ^{m}〈i∂_{x}〉^{n}φ(x)∥ _{L2} < ∞} is the weighted Sobolev space. Then we give the main term of the large time asymptotics of solutions in the sub critical case.

Original language | English |
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Pages (from-to) | 275-308 |

Number of pages | 34 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2005 Feb |

Externally published | Yes |

## Keywords

- Dissipative evolution equations
- Large time asymptotics
- Non-convective type
- Sub-critical case

## ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)