Asymptotics for the Ostrovsky-Hunter equation in the critical case
Summary: We consider the Cauchy problem for the Ostrovsky-Hunter equation \[\partial_x (\partial_t u - (b / 3) \partial_x^3 u - \partial_x \mathcal{K} u^3) = a u, \quad (t, x) \in \mathbb{R}^2, \quad u (0, x) = u_0(x), \ x \in \mathbb{R}, \] where \(a b > 0\). Define \(\xi_0 = (27 a / b)^{1 / 4}\). Suppose that \(\mathcal{K}\) is a pseudodifferential operator with a symbol \(\widehat{K} (\xi)\) such that \(\widehat{K} (\pm \xi_0) = 0\), \(\Im\widehat{K}(\xi) = 0\), and \(|\widehat{K} (\xi)| \leq C\). For example, we can take \(\widehat{K} (\xi) = (\xi^2 - \xi_0^2 )/ (\xi^2 + 1)\). We prove the global in time existence and the large time asymptotic behavior of solutions.
scientific article; zbMATH DE number 6915934
scientific article