I would like to study the well-posedness of the following equation


$u\_t - u\_{txx} + a u + b u\_x + c u\_{xx} + \gamma u\_{xxx} = f$


with $u(0)=0$ and $f \in H^{s-1}(\mathbb{R})$, where $a, b, c, \gamma$ are all uniformly bounded functions of $t, x$. What I want to have is that there is a unique solution $u\in H^{s}(\mathbb{R})$.


I tried to use the energy estimates but didn't figure it out. For simplicity, we can assume $s=4$.


Thank you.



A linear equation related to Camassa-Holm equation
Set $v=u-u\_{xx}$. Then your equation is of the form $v\_t=-\gamma v\_x+Av+f$, where $A$ is a bounded
operator. The rest is routine.


