Instantaneous point source solutions in nonlinear diffusion with nonlinear loss or gain

Abstract

AbstractExact solutions are developed for instantaneous point sources subject to nonlinear diffusion and loss or gain proportional to nth power of concentration, with n > 1. The solutions for the loss give, at large times, power-law decrease to zero of slug central concentration and logarithmic increase of slug semi-width. Those for gain give concentration decreasing initially, going through a minimum, and then increasing, with blow-up to infinite concentration in finite time. Slug semi-width increases with time to a finite maximum in finite time at a blow-up. Taken in conjunction with previous studies, these new results provide an overall schema for instantaneous nonlinear diffusion point sources with nonlinear loss or gain for the total range n ≥ 0. Six distinct regimes of behaviour of slug semi-width and concentration are identified, depending on the range of n, 0 ≤ n < 1, n = 1, or n > 1. Three of them are for loss, and three for gain. The classical Barenblatt-Pattle nonlinear instantaneous point-source solutions with material concentration occupy a central place in the total schema.