I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions:
$$\begin{cases}\dfrac{\partial u}{\partial t}-\Delta u=f(t,x), & (t,x)\in(0,T)\times \Omega\\ \dfrac{\partial u}{\partial\nu}=0, & (t,x)\in (0,T)\times\partial\Omega \\ u(0,x)=u_0(x), & x\in\Omega\end{cases}$$
where $\Omega\subset\mathbb{R}^N$ is a bounded, connected and open Lipschitz (not $C^2$) domain and $f\in L^2(0,T;L^2(\Omega)),\ u_0\in L^2(\Omega)$. My questions are:
1. Is $u\in C^{\alpha/2,\alpha}((0,T]\times\overline{\Omega})$, for some $\alpha\in (0,2)$? (Definition of parabolic Holder spaces: Lunardi, page 177)
2. Is $u\in C((0,T];L^{\infty}(\Omega))$?
3. Are there some newer works for the regularity of weak solutions of parabolic problems with maybe clearer statements? I am asking this because the elliptic regularity is very well writen in many monographs but for parabolic equations the books below are the only I know...
I know the book Ladyzenskaja- Linear and quasi linear equations of parabolic type (chapters III and IV). But here only Dirichlet boundary conditions are considered and the domain has smooth boundary - Theorem 10.1/page 204 - Holder continuity.
I also take a look at A. Lunardi - Analytic semigroups and optimal regularity in parabolic problems but there $u_0,f$ are always assumed to be more regular.
Then I see the book of Liebermann-Second order Parabolic differential equations which is newer but it seems that it makes more local estimates - however, looking page by page I didn't succeed to find something usable in my situation.
