Continuous dependence on the derivative of generalized heat equations. (English) Zbl 1349.35150

Franco, Tertuliano; Haddad, Julián

Differ. Integral Equ. 28, No. 1-2, 59-78 (2015).

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.

Reviewer: Elena Frolova (St. Petersburg)

MSC:

35K10	Second-order parabolic equations
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

generalized heat equation; one dimensional torus; measure; Fourier transform; continuous dependence