Well posedness of Burgers' equation

let’s consider the general Burgers’ equation

$$ frac{partial u}{partial t} + c(x) frac{partial u}{partial x} = nu frac{partial ^{2}u}{partial x^{2}} $$

where $c(x)$ is a periodic and bounded function and $v$ is a constant.

I want to state sufficient conditions such that the problem is well-posed. I know that a problem is well-posed if there exist a solution $u(x,t)$ for the initial condition $g(x)$ s.t
$$ ||u(cdot,t)||_{L_2} leq k e^{alpha t} ||g(cdot)||_{L_2}$$ Here I arbitrary chose the $L_2$ norm.

Given the definition, how can I state sufficient conditions on $c(x)$ and $v$ such that the Burgers’ equation is well-posed?