Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions

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We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball uₜ=Δuᵐ+a(x)uᵖ, for m>0, 0<p≤{1,m}, a(x)=1Bₗ(x), 0<L<∞ and N≥2. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length L plays a crucial role in the critical case p=m for N≥3. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when L=∞.