Consider the BBM equation: $-u_{txx}+u_{t}=u_{x},\quad u(x,0)=u_0(x),\quad x,t\in{\bf R}$
One may rewrite it as
$u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ exists.
Here are my questions:
- Does $(I-A)^{-1}$ always exist?
- Is there an integral operator $K:L^2({\bf R})\to L^2({\bf R})$ such that $K=((I-A)^{-1}\partial_x)$?