Continuous dependence on the derivative of generalized heat equations. (Q2341788)

The authors consider a generalized heat equation \(\partial_t \rho=\frac{d}{dx}\frac{d}{dW}\rho,\) where W is a finite measure on the one-dimensional torus, and \(\frac{d}{dW}\) is the Radon-Nikodym derivative with respect to \(W.\) Continuous dependence of the solution \(\rho\) as a function of \(W\) is obtained under the assumption that the Lebesgue measure is absolutely continuous with respect to \(W.\) The generalized heat equation is transformed into an equivalent classical partial differential equation. The equivalent version of the equation is investigated by the help of Fourier transform with respect to time. The paper contains several examples illustrating the range of applicability of the result.