Asymptotics in the critical case for Whitham type equations

Consider the Cauchy problem for nonlinear dissipative evolution equations {ut+N(u,u)+Lu=0,x∈R,t>0,u(0,x)=u0(x),x∈R, where LL is the linear pseudodifferential operator Lu=F¯ξ→x(L(ξ)û(ξ)) and the nonlinearity is a quadratic pseudodifferential operator N(u,u)=F¯ξ→x∫RA(t,ξ,y)û(t,ξ−y)û(t,y)dy,û≡Fx→ξu is direct Fourier transformation. Let the initial data u0∈Hβ,0∩H0,β, β>12, are sufficiently small and have a non-zero total mass M=∫u0(x)dx≠0, here Hn,m={ϕ∈L2‖〈x〉m〈i∂x〉nϕ(x)‖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 MM of the initial data.