TY - JOUR

T1 - Blow-up directions for quasilinear parabolic equations

AU - Seki, Yukihiro

AU - Umeda, Noriaki

AU - Suzuki, Ryuichi

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/4

Y1 - 2008/4

N2 - We consider the Cauchy problem for quasilinear parabolic equations u • = Δ φ (u) + f(u), with the bounded non-negative initial data u0(x) (u0(x) ≢ 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 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 ||u0|| • ∞ (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 u_0 for blow-up with the least blow-up time, provided that f(ξ) grows more rapidly than φ(ξ).

AB - We consider the Cauchy problem for quasilinear parabolic equations u • = Δ φ (u) + f(u), with the bounded non-negative initial data u0(x) (u0(x) ≢ 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 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 ||u0|| • ∞ (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 u_0 for blow-up with the least blow-up time, provided that f(ξ) grows more rapidly than φ(ξ).

UR - http://www.scopus.com/inward/record.url?scp=48849110801&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=48849110801&partnerID=8YFLogxK

U2 - 10.1017/S0308210506000801

DO - 10.1017/S0308210506000801

M3 - Article

AN - SCOPUS:48849110801

VL - 138

SP - 379

EP - 405

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 2

ER -