A nonnegative blowing up solution of the semilinear heat equation u t = ∆u + u p with p > 1 is considered when initial data u 0 satisfiesIt is shown that the solution blows up only at space infinity and that lim |x|→∞ u(x, t) is the solution of the ordinary differential equation v t = v p with v(0) = M.  2005 Elsevier Inc. All rights reserved.

We consider the Cauchy problem for quasilinear parabolic equations ut = ∆φ(u) + f (u), with the bounded non-negative initial data u 0 (We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation dv/dt = f (v) with the initial data u 0 L ∞ (R N ) > 0. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on u 0 for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on u 0 for blow-up with the least blow-up time, provided that f (ξ) grows more rapidly than φ(ξ).

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We discuss the motion of noncompact axisymmetric hypersurfaces Γt evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with that of the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.