Let $u_{t}+u_{xxx}=f,\,\, u(x,0)=0,\,x\in(0,1), \, t\in[0,T]$
$u(0,t)=0,u(1,t)=0, u_{x}(1,t)=0$.
Prove that \begin{equation} \boxed{\lVert u \rVert_{L^{2}(0,T;H^{2}(0,1))}\leq C\lVert f\rVert_{ L^{2}(0,T;L^{2}(0,1))}}. \end{equation} where C is a constant.
I already succeeded on proving $\int^{T}_{0}\int^{1}_{0}u^{2}(x,t)dxdt\leq \int^{T}_{0}\int^{1}_{0}f^{2}(x,t)dxdt$ and $\int^{T}_{0}\int^{1}_{0}u_{x}^{2}(x,t)dxdt\leq \int^{T}_{0}\int^{1}_{0}f^{2}(x,t)dxdt$.
Now I'm trying to prove particularly $\int^{T}_{0}\int^{1}_{0}u_{xx}^{2}(x,t)dxdt\leq \int^{T}_{0}\int^{1}_{0}f^{2}(x,t)dxdt$.
In order to prove this result, I multiplied the linear kdv $u_{t}+u_{xxx}=f$ both sides by $xu_{xx}$ and used integration by parts technique. But I couldn't succeed in this approach. Any hints or suggestion will be very much appreciated.
Thanks in advance.
