Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

Abstract

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation $\frac{{\partial u}} {{\partial t}} + a\frac{{u^{2} \partial u}} {{\partial x}} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$(x,t) ∈ R ². We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function u ₀(x) ∈ H ^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2.