Electron. J. Diff. Eqns., Vol. 2007(2007), No. 57, pp. 1-12.

Sufficient conditions for nonexistence of gradient blow-up for nonlinear parabolic equations

Aris S. Tersenov

In this paper we study the initial-boundary value problems for nonlinear parabolic equations without Bernstein-Nagumo condition. Sufficient conditions guaranteeing the nonexistence of gradient blow-up are formulated. In particular, we show that for a wide class of nonlinearities the Lipschitz continuity in the space variable together with the strict monotonicity with respect to the solution guarantee that gradient blow-up cannot occur at the boundary or in the interior of the domain.

Submitted February 8, 2006. Published April 17, 2007.
Math Subject Classifications: 35K55, 35K15, 35A05.
Key Words: Bernstein-Nagumo condition; gradient blow-up; a priori estimates nonlinear parabolic equation.

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Aris S. Tersenov
Department of Mathematics and Statistics
University of Cyprus
P.O. Box 20537, 1678 Nicosia, Cyprus
Tel.:+357 22892560, Fax:+357 22892550
email: aterseno@ucy.ac.cy

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