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 ut=Δϕ(u)+f(u)ut=Δϕ(u)+f(u) with initial data u0∈L∞(RN)u0∈L∞(RN), where ϕ and f   are nonnegative functions satisfying ϕ″⩽0ϕ″⩽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)v′=f(v) with initial data ‖u0‖L∞(RN)‖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.