Asymptotic behavior of solutions of diffusion-like partial differential equations invariant to a family of affine groups
L. Dresner · 1990
This report deals with the asymptotic behavior of certain solutions of partial differential equations in one dependent and two independent variables (call them c, z, and t, respectively). The partial differential equations are invariant to one-parameter families of one-parameter affine groups of the form: c{prime} = {lambda}{sup {alpha}}c, t{prime} = {lambda}{sup {beta}}t, z{prime} = {lambda}z, where {lambda} is the group parameter that labels the individual transformations and {alpha} and {beta} are parameters that label groups of the family. The parameters {alpha} and {beta} are connected by a linear relation, M{alpha} + N{beta} = L, where M, N, and L are numbers determined by the structure of the partial differential equation. It is shown that when L/M and N/M are <0, certain solutions become asymptotic to z{sup L/M}t{sup {minus}N/M} for large z or small t. Some practical applications of this result are discussed. 8 refs.