Multi-dimensional explosion due to a concentrated nonlinear source

Let D be a bounded n-dimensional domain, ∂D be its boundary, D¯ be its closure, T be a positive real number, B be an n-dimensional ball {x ∈ D: |x − b| 0, and ψ is nontrivial on ∂B, nonnegative, and continuous such that ψ = 0 on ∂D, ψxi is bounded for i = 1, 2, 3, . . . ,n, and Δψ + (∂χB(x)/∂ν)f (ψ(x)) 0 in D. It is shown that it has a unique solution u before a blow-up occurs. A criterion for u to blow up in a finite time is also given. If u exists in a finite time only, then u blows up somewhere on B¯.  2004 Elsevier Inc. All rights reserved