On directional blow-up for quasilinear parabolic equations with fast diffusion

We discuss blow-up at space infinity of solutions to quasilinear parabolic equations of the form u_t = Δ φ{symbol} (u) + f (u) with initial data u₀ ∈ L^∞ (R^N), where φ{symbol} and f are nonnegative functions satisfying φ{symbol}^″ ≤ 0 and ∫₁^∞ d ξ / f (ξ) < ∞. We study nonnegative blow-up solutions whose blow-up times coincide with those of solutions to the O.D.E. v^′ = f (v) with initial data {norm of matrix} u₀ {norm of matrix}_{L∞ (RN)}. We prove that such a solution blows up only at space infinity and possesses blow-up directions and that they are completely characterized by behavior of initial data. Moreover, necessary and sufficient conditions on initial data for blow-up at minimal blow-up time are also investigated.