Asymptotics of solutions for sub critical non-convective type equations

✍ Scribed by Nakao Hayashi; Elena I. Kaikina; Pavel I. Naumkin

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 258 KB
• Volume: 28
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.558

✦ Synopsis

Abstract

We study the Cauchy problem for non‐linear dissipative evolution equations
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where ℒ︁ is the linear pseudodifferential operator and the non‐linearity is a quadratic pseudodifferential operator
equation image
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 $u_{0} \in {\bf H}^{\varrho,0} \cap {\bf H}^{0, \varrho}, \varrho > {1\over 2}$, are sufficiently small and have a non‐zero total mass $\int u_{0}(x){\rm d}x \ne 0$, where is the weighted Sobolev space. Then we give the main term of the large time asymptotics of solutions in the sub critical case. Copyright © 2004 John Wiley & Sons, Ltd.