Blow-up of solutions of a quasilinear parabolic equation

Abstract

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations u_t = Δu^m + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up condition

We show that if ξ^m+2/N /(−log ξ)^β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limt_t→∞ inf_|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.