Asymptotics of solutions for sub critical non-convective type equations

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) = u₀(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→ξu 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 u₀ ∈ H ^q,0 ∩ H^{0, q}, q> 1/2, are sufficiently small and have a non-zero total mass ∫ u₀(x)dx ≠ 0, where H ^n,m = {φ ∈ L²∥〈x〉 ^m〈i∂_x〉ⁿφ(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.