Let alpha, b and T be positive numbers, D = (0, infinity), <(Dover bar> = [0, infinity), and Omega = D x (0, T]. This article studies the first initial-boundary value problem, u(t) - u(xx) = alpha delta(x - b)f (u(x, t)) in Omega, u(x, 0) = psi(x) on <(Dover bar>, u(0, t) = 0 = lim(x ->infinity) u(x, t) for 0 < t <= T, where delta(x) is the Dirac delta function, and f and psi are given functions. We assume that f (0) >= 0, f(u) and its derivatives f'(u) and f ''(u) are positive for u > 0, and psi(x) is nontrivial, nonnegative and continuous such that psi(0) = 0 = lim(x ->infinity) psi(x), and psi '' + alpha delta(x - b)f (psi) >= 0 in D. It is shown that if u blows up, then it blows up in a finite time at the single point b only. A criterion for u to blow up in a finite time and a criterion for u to exist globally are given. It is also shown that there exists a critical position b* for the nonlinear source to be placed such that no blowup occurs for b <= b*, and u blows up in a finite time for b > b*. This also implies that u does not blow up in infinite time. The formula for computing V is also derived. For illustrations, two examples are given.