Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

Abstract EN

We consider the Cauchy problem for the Ostrovsky-Hunter equation ∂ x ∂ t u - b / 3 ∂ x 3 u - ∂ x K u 3 = a u , t , x ∈ R 2 , u 0 , x = u 0 x , x ∈ R , where a b > 0 .

Define ξ 0 = 27 a / b 1 / 4 .

Suppose that K is a pseudodifferential operator with a symbol K ^ ξ such that K ^ ± ξ 0 = 0 , I m K ^ ξ = 0 , and K ^ ξ ≤ C .

For example, we can take K ^ ξ = ξ 2 - ξ 0 2 / ξ 2 + 1 .

We prove the global in time existence and the large time asymptotic behavior of solutions.