Blow-up directions for quasilinear parabolic equations

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).