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| < R} centered at b ∈ D with a radius R, B be its closure, ∂B be its boundary, ν denote the unit inward normal at x ∈ ∂B, and χ B (x) be the characteristic function. This article studies the following multi-dimensional parabolic first initialboundary value problem with a concentrated nonlinear source occupyingB:where the normal vector ν(x) is extended to a vector field defined in the whole domain D, f and ψ are given functions such that f = 0 for x / ∈B, f (0) 0, f (0) 0, f (u) and f (u) are positive for u > 0, and ψ is nontrivial on ∂B, nonnegative, and continuous such that ψ = 0 on ∂D, ψ x i is bounded for i = 1, 2, 3, . . . , n, and ∆ψ + (∂χ B (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 onB.  2004 Elsevier Inc. All rights reserved.

Abstract.Let q be a nonnegative real number, and T be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem:where S(x) is the Dirac delta function, and / and ip are given functions. It is shown that the problem has a unique solution before a blow-up occurs, u blows up in a finite time, and the blow-up set consists of the single point b. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.

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