Grow-up for a quasilinear heat equation with a localized reaction

We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized u(t) = (u(m))xx + a(x)u(p), m, p > 0 and a(x) being the characteristic function of an interval. We prove that there exists an exponent po = max{1, 11+1} such that all global solutions are bounded if p > pp, while for p < pp all the solutions are global and unbounded. In the last case, we prove that if p < m the grow-up rate is different to the one obtained when a(x) 1, while if p > m the grow-up rate coincides with that rate, but only inside the support of a; outside the interval the rate is smaller. (C) 2019 Elsevier Inc. All rights reserved.