“Waiting-time” Solutions of a Nonlinear Diffusion Equation

Similarity solutions of the equation $\partial c/\partial t = ( \partial /\partial x ) ( c^n \partial c/\partial x ),n > 0$, are studied for both $t > 0$ and $t < 0$. A two-parameter family of weak solutions of the equation is constructed which for $t < 0$ have support in $x < 0$, i.e., $c = 0$ for $x > 0$ and there is a stationary free boundary at $x = 0$, and for $t > 0$ have support in $x < at^\beta $ where $\beta < 1,a < 0$, i.e., there is a moving free boundary $x = at^\beta $. By prescribing appropriate “initial” data $c = c_0 ( x ),x < 0$, at $t = - t_0 $, these solutions demonstrate the existence of a “waiting-time” $t_0 $ before the free boundary moves. A comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of “waiting-times” for general initial data. A three parameter family of solutions which exhibit “blow-up” at $t = 0$ is also constructed. Applications of these results to models for the spreading of a viscous blob on a horizontal plane surface and the expansion of a region of water vapor in a dry powder are suggested.