Compatible initial condition for the numerical scheme of PDE

Question

I have to solve numerically the following PDE ($0<t_0<t$): $$\partial_t H(x,t,t_0) = \partial_x^2 H(x,t,t_0)+(1-2U(x,t))H(x,t,t_0)$$ with the initial condition: $$H(x,t=t_0,t_0) = U^2(x,t_0)$$ The numerical scheme I use is implicit, $$\frac{H^{n+1}_{k}-H^{n}_{k}}{dt} = \frac{H^{n+1}_{k+1}-2H^{n+1}_{k}+H^{n+1}_{k-1}}{dx^2} + (1-2U^{n+1}_{k})H^{n+1}_{k}$$ where U is a function which I know the numerical solution at every time step from 0 to t.

Question: What should be the initial condition to be used for this scheme?

1. $ H^0_k = (U^0_k)^2 $ ?
2. $ H^0_k = (U^0_{k+1})^2 $ ?
3. $ H^0_k = U^0_{k+1} U^0_k $ ?

Indeed, I tried the first one, but the result seems not correct.

P/s: $dx = 0.5$ and $dt = 0.1$