Continuous dependence on the derivative of generalized heat equations

We consider here a generalized heat equation $\partial_t ρ=\frac{d}{dx}\frac{d}{dW}ρ$, where $W$ is a finite measure on the one dimensional torus, and $\frac{d}{dW}$ is the Radon-Nikodym derivative with respect to $W$. Such equation has appeared in different contexts, being related to physical systems and representing a large class of classical and non-classical parabolic equations. As a natural assumption on $W$, we require that the Lebesgue measure is absolutely continuous with respect to $W$. The main result here presented consists in proving, for a suitable topology, a continuous dependence of the solution $ρ$ as a function of $W$.

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