Single blow-up point and critical speed for a parabolic problem with a moving nonlinear source on a semi-infinite interval

Abstract

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,where δ (x) is the Dirac delta function, and f and ψ are given functions. It is shown that if the solution u blows up in a finite time t b , then it blows up only at the point x = vt b . A criterion for u to exist globally and a criterion for u to blow up in a finite time are given. Furthermore, the problem is shown to have a critical speed v * of the moving nonlinear source such that no blowup occurs for v ≥ v * and blowup occurs in a finite time for v < v * . The formula for computing v * is also derived.