2017
Том 69
№ 9

All Issues

Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds

Antoniouk A. Val., Antoniouk A. Vict.

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Abstract

We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time. This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.

Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold. Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation.

English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 11, pp 1632-1652.

Citation Example: Antoniouk A. Val., Antoniouk A. Vict. Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds // Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1454–1472.

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