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Global blow-up for a semilinear heat equation on a subspace

We study the asymptotic behaviour as t → T– , near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term: with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T– , revealing a non-uniform global blow-up: uniformly on any compact set [δ, 1], δ ∈ (0, 1).