Mixed boundary condition of parabolic equations

Question

Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that $$ \partial\Omega=\partial\Omega_D\cup\partial\Omega_N, $$ where $ \partial\Omega_D $ and $ \partial\Omega_N $ are nonempty smooth and connected. Consider the initial boundary problem as follows. $$ \left\{\begin{aligned} &\partial_tu-\Delta u=F&\text{ in }&\Omega\times(0,T),\\ &u(x,0)=u_0(x)&\text{ on }&\Omega\times\{0\},\\ &u(x,t)=0&\text{ on }&\partial\Omega_D\times(0,T),\\ &\frac{\partial u}{\partial\nu}(x,t)=0&\text{ on }&\partial\Omega_N\times(0,T), \end{aligned}\right. $$ where $ \nu $ is the outer unit normal vector of $ \partial\Omega $ and $ F\in L^p(\Omega\times(0,T)) $. The above model is of mixed boundary conditions. I wonder if I can get the regularity estimates for $ u $. Can you give me some hints or references?