I am dealing with the following partial differential equation.
\begin{equation} \left\{ \begin{array}{lll} u_t & = \nabla \cdot \left(\frac{\nabla u}{1+w^2}\right) & \text{ in } Q_T; \\ -\epsilon \Delta w & = \bigl|\nabla f\bigl|^2 - w & \text{ in } Q_T; \\ \frac{\partial u}{\partial n}(x,t) & =0 & \text{ on } \partial \Omega \times (0, T); \\ w(x,t) & =0 & \text{ on } \partial \Omega \times (0, T); \\ u(x,0) & =u_0(x) & \text{ in } \Omega \ \end{array} \right. \end{equation} with $\Omega = \{x \in R^2 | ||x||_2<1\}$ and $Q_T=\Omega \times (0,T)$. I have $f \in H^2(\Omega)$ and therefore there exist a weak solution $w \in H_0^1(\Omega)$ and we can even proof that $w \in H^3(\Omega)$.
Moreover, we obtain the following inequality by Sobolev embedding theorems. $$||w||_{C^1(\bar{\Omega})}\leq C(\varepsilon)||f||_{H^3(\Omega)}$$ for all $t \in (0,T_0)$. I can therefore proof that there exist a weak solution of the partial differential equation in $L^2(H^1(\Omega);(0,T_0))$. So we have $u \in H^1(\Omega)$ in the space an $u \in L^2(0,T_0)$ in time.
Now i need for further calculations that $u \in H^3(\Omega)$. Is it possible that i get this higher regularity by a bootstrapping argument? And if this is the case, how exactly does it work? I just got the hint and read through some papers on bootstrapping but didn't really figure it out.