Explicit solutions are obtained for evolution equations for explosively unstable situations. These solutions include the effects of diffusion with linear or quadratic density dependence of the diffusion coefficient. As a result of balance between the diffusion and nonlinear terms, explosive growth in time can occur with a preservation in shape of certain spatial distributions. The solutions are generalized to cases of two interacting populations.

• Received 21 January 1987

DOI:https://doi.org/10.1103/PhysRevA.36.965