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

discuss blow-up at space infinity of solutions to quasilinear parabolic equations of the form ut = φ(u) + f (u) with initial data u0 ∈ L∞(RN), where φ and f are nonnegative functions satisfying φ 0 and ∞1 dξ/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 u0 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.