I am trying to solve the following PDE. I suspect it is a transport equation but I am not sure.
$$ \partial_t u(x,t) + \partial_x\left[f(x,t)u(x,t)\right] = -\left(\dfrac{\phi_1}{f(x,t)} + \dfrac{\phi_2}{g(x,t)}\right) $$
where $\partial_t u(x,t)$ denotes the partial derivative with respect to time and $\partial_x$ represents the derivative with respect to $x$. I am trying to solve for the function $u(x,t)$. $\phi_1$ and $\phi_2$ are just parameters.
How can I solve this equation analytically? Does it have a name?
P.S.: I am thinking of using the following initial condition.
$$u(x,0) = \dfrac{c}{f(x,0)}exp\left(-\int \left(\dfrac{\phi_1}{f(x,0)}+\dfrac{\phi_2}{g(x,0)}\right)dx\right)$$
If this is not a possibility, can somebody illustrate how this would work with a simpler initial condition or provide a sketch of the general solution?