Blow-Up Solutions of the Two-Dimensional Heat Equation Due to a Localized Moving Source

The problem examined is that of a localized energy source which undergoes planar motion along the surface of a reactive-diffusive medium. This is representative of a laser beam that is moving across the flat surface of a combustible material during a cutting, welding or heat treating process. The mathematical model for this situation is a heat equation in two-dimensions with a nonlinear source term, which is localized around a reference point that is allowed to move. Results are derived that indicate the roles played by the size, strength and motion of the localized source in determining whether or not a blow-up occurs.