Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

Abstract

We consider the Cauchy problem for the Ostrovsky-Hunter equation ${\partial }_x\left({\partial }_{t}u-\left(b/\mathrm{3}\right){\partial }_x^{\mathrm{3}}u-{\partial }_x\mathcal{K}{u}^{\mathrm{3}}\right)=au$, $\left(t,x\right)\in {\mathbb{R}}^{\mathrm{2}}$, $u\left(\mathrm{0},x\right)=u_{\mathrm{0}}\left(x\right)$, $x\in \mathbb{R}$, where $ab>\mathrm{0}$. Define ${\xi }_{\mathrm{0}}={\left(\mathrm{27}a/b\right)}^{\mathrm{1}/\mathrm{4}}$. Suppose that $\mathcal{K}$ is a pseudodifferential operator with a symbol $\widehat{K}\left(\xi \right)$ such that $\widehat{K}\left(\pm {\xi }_{\mathrm{0}}\right)=\mathrm{0}$, $\mathrm{I}\mathrm{m} \widehat{K}\left(\xi \right)=\mathrm{0}$, and $\left|\widehat{K}\left(\xi \right)\right|\le C$. For example, we can take $\widehat{K}\left(\xi \right)=\left({\xi }^{\mathrm{2}}-{\xi }_{\mathrm{0}}^{\mathrm{2}}\right)/\left({\xi }^{\mathrm{2}}+\mathrm{1}\right)$. We prove the global in time existence and the large time asymptotic behavior of solutions.