I am finding an analytic expression for the solution of the transport PDE:

$$u_t+left(frac{1-2u(x,t)}{a}right)u_x = 0,quad a=const, quad x>0, quad t >0$$

$$u(x=0,t) = u_0, quad u_0 in (0,1)$$

$$u(x,t=0) = varphi(x)$$

I have got the solution:

$$

u(x,t) = varphileft(x-frac{1-2u(x,t)}{a},tright)

$$

My question is how to use boundary condition $u_0$? I have checked many different books and articles, but stuck though.

Thank you.