Asymptotics in the critical case for Whitham type equations

Consider the Cauchy problem for nonlinear dissipative evolution equations {(u_t + N (u, u) + L u = 0,, x ∈ R, t > 0,; u (0, x) = u₀ (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 u₀ ∈ H^{β, 0} ∩ H^{0, β}, β > frac(1, 2), are sufficiently small and have a non-zero total mass M = ∫ u₀ (x) d x ≠ 0, here H^{n, m} = {φ{symbol} ∈ L² {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.