Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

Abstract

Let α, b and T be positive numbers, D = (0, ∞),D = [0, ∞), and Ω = D × (0, T ]. This article studies the first initial-boundary value problem, u t − u xx = αδ(x − b)f (u(x, t)) in Ω, u(x, 0) = ψ(x) onD, u(0, t) = 0 = lim x→∞ u(x, t) for 0 < t ≤ T, where δ (x) is the Dirac delta function, and f and ψ 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 ψ(x) is nontrivial, nonnegative and continuous such that ψ (0) = 0 = lim x→∞ ψ (x), and ψ + αδ(x − b)f (ψ) ≥ 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 b * is also derived. For illustrations, two examples are given.