Blow-up of solutions of a quasilinear parabolic equation

Abstract

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = ∆u m + f (u), where m > 1 and f (ξ) is a positive function in ξ > 0 satisfying f (0) = 0 and a blow-up conditionWe 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 u 0 having compact support exists globally in time and grows up to ∞ as t → ∞: limt→∞ inf |x|<R u(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.