Results for heat equation with interior point condition

Question

I am looking for any results concerning the study (and for example potential existence and uniqueness results) of the heat equation with Dirichlet boundary conditions and with an additional interior point condition, i.e: $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},x\in[0,1],t\geq 0$$ with $u(0,t)=u(z,t)=u(1,t)=0$, $z\in(0,1)$, and $u(x,0)=f(x)$ for all $x\in[0,1]$. I was wondering if the classical existence and uniqueness results were extendable to this case, and whether this could be done in an obvious way? Any comments or references to literature would be greatly appreciated.