Noriaki Umeda

Blow-up directions for quasilinear parabolic equations

We consider the Cauchy problem for quasilinear parabolic equations ut = ∆φ(u) + f(u) with the bounded nonnegative initial data u0(x) ( 6≡ 0), where f(ξ) is a positive function in ξ > 0 satisfying a blow-up condition ∫ ∞ 1 1/f(ξ) dξ < ∞. We study blow-up nonnegative 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 ‖u0‖L∞(RN ) > 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 u0 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 u0 for blow-up with the least blow-up time, provided that f(ξ) grows up more rapidly than φ(ξ). The first and third authors were supported by The 21st Century COE Program “New Mathematical Development Center to Support Scientific Technology”, Japan Society for the Promotion of Science (JSPS). The second author was partially supported by Grant-in-Aid for Scientific Research (C), No. 17540171, JSPS. 1 UTMS 2006–20 August 18, 2006 Blow-up directions for quasilinear parabolic equations by Yukihiro Seki, Ryuichi Suzuki and Noriaki Umeda UNIVERSITY OF TOKYO GRADUATE SCHOOL OF MATHEMATICAL SCIENCES KOMABA, TOKYO, JAPAN

Blow-up at space infinity for nonlinear heat equations

The last condition of (A1) forces f to grow superlinearly at infinity. A nonlinear evolution equation may have a unique local-in-time solution in a suitable function space and it can be extend as a solution together with evolution of time so long as it belongs to the function space. However, in general, the Cauchy problem is not solvable globally in time; a solution may blow up in finite time. That is, there may exist a finite time T < ∞ such that the solution ceases to exist in the function space at the time T . This phenomenon is called blow-up in finite time and we call such a time T blow-up time. The Cauchy problem (1.1) has a unique local-in-time solution u = u(·, t) in L∞(RN ) for any nonnegative initial data u0 ∈ L∞(RN ). However, it may blow up in finite time. For instance, if the initial data does not decrease at space infinity, the solution of (1.1) does blow up in finite time. We are interested in the blow-up times of solutions and detailed behavior of solutions at the blow-up times. In particular, we discuss solutions which blow up at space infinity as we will state later.

Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

We discuss the motion of noncompact axisymmetric hypersurfaces Γ t evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with 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.

Blow-up of solutions of a quasilinear parabolic equation

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations u t = Δ u m + f(u) , where m > 1 and f (ξ) is a positive function in ξ > 0 satisfying f (0) = 0 and a blow-up condition We show that if ξ m +2/ N /(−log ξ) β = O ( f (ξ)) as ξ ↓ 0 for some 0 mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u 0 having compact support exists globally in time and grows up to ∞ as t → ∞: limt t →∞ inf | x | R u(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.