Asymptotics of solutions for sub critical non‐convective type equations

Abstract

AbstractWe study the Cauchy problem for non‐linear dissipative evolution equations equation image where ℒ︁ is the linear pseudodifferential operator and the non‐linearity is a quadratic pseudodifferential operator equation image û ≡ ℱx→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.