The Cauchy Problem for the Sixth Order $p$-Generalized Benney-Luke Equation

Abstract

We investigate the Cauchy problem for the sixth order $p$-generalized Benney-Luke equation. The local well-posedness is established in the energy space $\dot{H}^1 (\mathbb{R}^n)∩ \dot{H}^3(\mathbb{R}^n)$ for $1 ≤ n ≤ 10,$ by means of the Sobolev multiplication law and the contraction mapping principle. Moreover, we establish the energy identity of solutions and provide the sufficient conditions of the global existence of solutions by analyzing the properties of the energy functional.