A multi-dimensional quenching problem due to a concentrated nonlinear source in RN

Let B be a N-dimensional ball {x is an element of R-N : \X\ < R} centered at the origin with a radius R, <(B)over bar> be its closure, and partial derivative B be its boundary. Also, let v(x) denote the unit inward normal at x is an element of partial derivative B, and chi(B)(x) be the characteristic function, which is 1 for x is an element of B, and 0 for x is an element of R-N\B. This article studies the following multi-dimensional semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source on partial derivative B: u(t) - Delta u = alpha partial derivative chi(B)(x)/partial derivative vf(u) in R-N x (0, T], u(x, 0) = 0 for x is an element of R-N, u((x,t) --> 0 as \x\ --> infinity for 0 < t <= T, where alpha and T are positive numbers, f is a given function such that lim(a-->c-) f(u) = infinity for some positive constant c, and f (u) and its derivatives f'(u) and f"(u) are positive for 0 <= u < c. It is shown that the problem has a unique nonnegative continuous solution before quenching, occurs, and if it quenches in a finite time, then it quenches everywhere on OB only. It is proved that it always quenches in a finite time for N <= 2. For N >= 3, it is shown that there exists a unique number alpha* such that it exists globally for alpha <= alpha* and quenches in a finite time for alpha > alpha*. Thus, quenching does not occur in infinite time. A formula for computing alpha* is given. A computational method for finding the quenching time is devised. (C) 2007 Elsevier Ltd. All rights reserved.