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Blow-up directions for quasilinear parabolic equations. (English) Zbl 1167.35393
Summary: We consider the Cauchy problem for quasilinear parabolic equations u t =Δϕ(u)+f(u), with the bounded non-negative initial data u 0 (x)(u 0 (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 u 0 L ( N ) >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 u 0 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).
MSC:
35K55Nonlinear parabolic equations
35K15Second order parabolic equations, initial value problems
35K65Parabolic equations of degenerate type
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)