Asymptotics in the critical case for Whitham type equations Original Research Article

Consider the Cauchy problem for nonlinear dissipative evolution equations View the MathML source{ut+N(u,u)+Lu=0,x∈R,t>0,u(0,x)=u0(x),x∈R, Turn MathJax on where LL is the linear pseudodifferential operator View the MathML sourceLu=F¯ξ→x(L(ξ)û(ξ)) and the nonlinearity is a quadratic pseudodifferential operator View the MathML sourceN(u,u)=F¯ξ→x∫RA(t,ξ,y)û(t,ξ−y)û(t,y)dy, Turn MathJax on View the MathML sourceû≡Fx→ξu is direct Fourier transformation. Let the initial data View the MathML sourceu0∈Hβ,0∩H0,β, View the MathML sourceβ>12, are sufficiently small and have a non-zero total mass View the MathML sourceM=∫u0(x)dx≠0, here View the MathML sourceHn,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.