Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds

  • A. Val. Antoniouk
  • A. Vict. Antoniouk


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.
How to Cite
Antoniouk, A. V., and A. V. Antoniouk. “Nonexplosion and Solvability of Nonlinear Diffusion Equations on Noncompact Manifolds”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, no. 11, Nov. 2007, pp. 1454–1472,
Research articles