Question

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.

Answer 1

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.