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

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We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized uₜ=(uᵐ)xx+a(x) uᵖ, m, p>0 and a(x) being the characteristic function of an interval. we prove that there exists p₀={1,m+12} such that all global solution are bounded if p>p₀, while for p≤ p₀ all the solution 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.