EXISTENCE, UNIQUENESS AND QUENCHING FOR A PARABOLIC PROBLEM WITH A MOVING NONLINEAR SOURCE ON A SEMI-INFINITE INTERVAL

Let v and T be positive numbers, D = (0, ∞), Ω = D × (0, T ], and D¯ be the closure of D. This article studies the first initial-boundary value problem, ut − uxx = δ(x − vt)f (u(x, t)) in Ω, u(x, 0) = 0 on D, ¯ u(0, t) = 0, u(x, t) → 0 as x → ∞ for 0 < t ≤ T, where δ (x) is the Dirac delta function, and f is a given function such that limu→c− f(u) = ∞ for some positive constant c. It is shown that the problem has a unique nonnegative continuous solution u, and u(vt, t) is a strictly increasing function of t; also, if u exists for t ∈ [0, tq) with tq < ∞, then sup {u (x, t) : 0 ≤ x < ∞} reaches c − at tq.