NONAUTONOMOUS KOLMOGOROV PARABOLIC EQUATIONS WITH UNBOUNDED COEFFICIENTS

By Markus Kunze, Luca Lorenzi, A. Lunardi. 2008

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We study a class of elliptic operators A with unbounded coeffi- cients defined in I × R d for some unbounded interval IR. We prove that, for any s 2 I, the Cauchy problem u(s,·) = f 2 Cb(R d ) for the parabolic equation Dtu = Au admits a unique bounded classical solution u. This allows to associate an evolution family {G(t, s)} with A, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function G(t, s)f. Under suitable assumptions, we show that there exists an evolution system of measures for {G(t, s)} and we study the first properties of the extension of G(t, s) to the L p -spaces with respect to such measures.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/80b8197ea2b982c8ff6b3e8b888c53e088dfd3f8