Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

• Date: 2008
• Main Authors: Antoniouk, A.Val., Antoniouk, A.Vict.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2008
• Series: Український математичний журнал
• Subjects: Статті
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/164759
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2008. — Т. 60, № 10. — С. 1299–1316. — Бібліогр.: 14 назв. — англ.

We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of manifold there are estimates exponential in time on the continuity of diffusion process with respect to the initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to the initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent space, nor uses embeddings of manifold to linear spaces of higher dimensions.