Why is the maximum principle not valid here ?
$u:\mathbf R^n_{+}\times (0,1]$ and $u(x,t)=1-\frac{1}{\left(4\pi (t-1/2)\right)^{n/2}}e^{\frac{-|x+1|^2}{4t-2}}$ then $u$ is bounded by $1$ and satisfies $u_t-\Delta u=0$, for example $u(x,1/2)=1$ in the interior but $u$ is not constant
Why does the maximum principle not hold ?
EDIT: I have the solution here, but it is not mentioned where the maximum principle fails
