Asymptotics in the critical case for Whitham type equations

Abstract

Consider the Cauchy problem for nonlinear dissipative evolution equations { u t + N ( u , u ) + L u = 0 , x ∈ R , t > 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ R , where L is the linear pseudodifferential operator L u = F ¯ ξ → x ( L ( ξ ) u ̂ ( ξ ) ) and the nonlinearity is a quadratic pseudodifferential operator N ( u , u ) = F ¯ ξ → x ∫ R A ( t , ξ , y ) u ̂ ( t , ξ − y ) u ̂ ( t , y ) d y , u ̂ ≡ F x → ξ u is direct Fourier transformation. Let the initial data u 0 ∈ H β , 0 ∩ H 0 , β , β > 1 2 , are sufficiently small and have a non-zero total mass M = ∫ u 0 ( x ) d x ≠ 0 , here H n , m = { ϕ ∈ L 2 ‖ 〈 x 〉 m 〈 i ∂ x 〉 n ϕ ( x ) ‖ L 2 < ∞ } 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.