The initial value problem for a KdV-type equation and a related bilinear estimate

In this work we consider a certain nonlinear evolution partial differential equation, which is a fifth order modification of the Camassa-Holm equation. We will show that the Cauchy problem is locally well-posed for initial data, of arbitrary size, in the Sobolev space Hs($R$) for any s > 1/4. That is, we will prove existence of a local solution, uniqueness, and continuous dependence on the t-variable. We will show further that the Cauchy problem is globally well-posed for initial data in H1($R$).; We will also consider a variation of this equation—a PDE with the same linear terms and slightly different nonlinear terms. For this equation we will show that the Cauchy problem is locally well-posed for initial data in the Sobolev space Hs($R$) for s ≥ 1/4.; Both of these PDEs can be written as non-local Korteweg deVries type equations. For this reason, the methods we will use to prove well-posedness are similar to those used for the KdV equation. In particular, we will first convert the initial value problem to an integral equation; then we will prove that the integral equation has a solution, by means of appropriate bilinear estimates obtained using Fourier analysis techniques.