Let B be a N -dimensional ball {x โ R N : |x| < R} centered at the origin with a radius R, B be its closure, and โ B be its boundary. Also, let ฮฝ(x) denote the unit inward normal at x โ โ B, and ฯ B (x) be the characteristic function, which is 1 for x โ B, and 0 for x โ R N \B. This article studies the following multi-dimensional semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source on โ B:
where ฮฑ and T are positive numbers, f is a given function such that lim uโc -f (u) = โ for some positive constant c, and f (u) and its derivatives f (u) and f (u) are positive for 0 โค u < c. It is shown that the problem has a unique nonnegative continuous solution before quenching occurs, and if u quenches in a finite time, then it quenches everywhere on โ B only. It is proved that u always quenches in a finite time for N โค 2. For N โฅ 3, it is shown that there exists a unique number ฮฑ * such that u exists globally for ฮฑ โค ฮฑ * and quenches in a finite time for ฮฑ > ฮฑ * . Thus, quenching does not occur in infinite time. A formula for computing ฮฑ * is given. A computational method for finding the quenching time is devised.