Solving the PDE: $\frac{\partial\phi}{\partial t}=\frac{\partial^2\phi}{\partial x^2}-\phi f$?

Question

I would like find a solution $\{\phi(x,t);t\in\mathbb{R}^+,x\in\mathbb{R}\}$ of the following PDE:

$$ \frac{\partial\phi}{\partial t}=\frac{\partial^2\phi}{\partial x^2}-\phi f(x) , $$ with boundary conditions $$ \phi(0,x)=1,\text{ and }\frac{\partial\phi}{\partial t}(0,x)=f(x). $$ The function $f$ will ideally be as general as possible. If such generality is unhelpful, then I am willing to consider any particular choice of $f$ satisfying:

1. $f(x)=f(-x)$ for any $x\in\mathbb{R}$,
2. $f>0$, and
3. $f(x)<f(y)$ for any $|x|>|y|$.

For example, $f(x)=1/(1+x^2)$ or $f(x)=1/(e^x+e^{-x})$.

This seemingly simple looking problem is completely baffling me. Does anyone have any suggestion for how to proceed? Do we even have existence and uniqueness?