Global Well-posedness of the Nonhomogeneous Initial Boundary Value Problem for the Hirota Equation Posed in a Finite Domain

Abstract

We study a system described by a type of initial and boundary value problem of the Hirota equation with nonhomogeneous boundary conditions posed on a bounded interval. Firstly, we prove the local well-posedness of the system in the space \(H^s(0,1)\) by using an explicit solution formula and contraction mapping principle for any \(s\ge 1\). Secondly, we obtain the global well-posedness in \(H^1(0,1)\) and \(H^2(0,1)\) by the norm estimation. Especially, the main difficulty is that the characteristic equation corresponding to Hirota equation needs to be solved by construction and that nonlinear terms are taken into consideration. In addition, the norm estimate for global well-posedness of solution in \(H^1(0,1)\) and \(H^2(0,1)\) are complicated.