Solving a partial differential equation with initial value $0$ but a boundary condition with non-zero space derivative

Question

I have a function $f(x,t)$ given by: $$f(x,t)=\cases{0 , & $x \le x_c$ \\ f_2(x,t), & $x>x_c$}$$ of which $f_2(x)$ satisfies the following partial differential equation: $$\frac{\partial f_2}{\partial t}=-c\frac{\partial f_2}{\partial x}+D\frac{\partial^2f_2}{\partial x^2}$$

where $c,D$ are constants with boundary conditions:

1. $f_2(x,0)=0$
2. $f_2(\infty,t)=0$
3. $\frac{\partial f_2}{\partial x}=\sigma(t) \,\,\,\,\, \text{at} \,\,\, x=x_c$

Mathematically, due to the initial condition $1$, it seems that $f_2=0$ will be the solution. But intuition says that due to condition $3$, $f_2$ will have a non-zero evolution with time.

How to solve this?