Assume that one has a classical PDE, say: $u_t(t,x)-u_{xx}(t,x)=0$, $t\in (0,1)$, $x\in (0,2)$, and $u(0,x)=0$. Then we can prove existence (and uniqueness) of solution when boundary conditions: $u(t,0)=f(t)$, $u(t,2)=g(t)$ are prescribed.
My question: what if one replaces the first bc by $u(t,\color{red}{1})=f(t)$? How can we prove existence in this case?
Is there any way to get back to the usual bcs?
Edit: What if one considers instead $u(t,0)=u(t,\color{red}{1})=f(t)$ while keeping the second bc at $x=2$?