Nonlinear heat diffusion under impulsive forcing

A nonlinear diffusion equation under the influence of an external forcing of distribution type is considered. The initial value problem on the line is shown to be equivalent to two initial/boundary value (IBV) problems on the semiline connected through a “jump” condition on the Neumann data at the origin. The two IBV problems for the nonlinear diffusion equation are mapped into two IBV problems for the linear heat equation, characterized by moving boundaries. The motion of the boundaries is a priori unknown and has to be determined as part of the solution. The moving boundary problems for the linear heat equation are reduced to a nonlinear Volterra integral equation which is shown to admit a unique solution for small intervals of time.