EFFECTS OF A CONCENTRATED NONLINEAR SOURCE ON QUENCHING IN R N

. Let T and α be positive real numbers, β be a real number, B be a N-dimensional ball  x ∈ R N : |x| < R centered at the origin with a radius R, and ∂B be its boundary. Also, let ν(x) denote the unit inward normal at x ∈ ∂B, and χB(x) be the characteristic function, which is 1 for x ∈ B, and 0 for x ∈ R N \ B. This article studies the following parabolic Cauchy problem with a concentrated nonlinear source on ∂B: ut − 4u = α(1 + |x|) β ∂χB(x) ∂ν f(u) in R N × (0, T], u(x, 0) = 0 for x ∈ R N , u(x, t) → 0 as |x| → ∞ for 0 < t ≤ T, where f is a given function such that limu→c− f(u) = ∞ for some positive constant c, and f(u) and its derivatives f 0 (u) and f 00(u) are positive for 0 ≤ u < c. It is shown that the solution u always quenches for N ≤ 2, and quenching can be prevented for any β for N ≥ 3. For given R and β, the effects of α on quenching are discussed. Similarly for a given α, the effects of R and β on quenching are investigated.