In this paper, the hyperbolic-parabolic mixed type equation \begin{document}$\frac{\partial u}{\partial t} = Δ A(u)+\text{div}(b(u)),\ \ (x,t)∈ Ω × (0,T),$ \end{document} with the homogeneous boundary condition is considered. We find that only a part of the boundary condition is able to ensure the posedness of the solutions. By introducing a new kind of entropy solution matching the part boundary condition in a special way, we obtain the existence of the solution by the $BV$ estimate method, and establish the stability of the solutions by the Kruzkov bi-variables method.