Exact sharp-fronted solutions for nonlinear diffusion on evolving domains

Abstract

Abstract Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either report numerical solutions or require an assumption of linear diffusion to determine exact solutions. Unfortunately, numerical solutions do not reveal the relationship between the model parameters and the solution features. Additionally, experimental observations typically report the presence of sharp fronts, which are not captured by linear diffusion. Here we address both limitations by presenting exact sharp-fronted solutions to a model of degenerate nonlinear diffusion on a growing domain. We obtain the solution by identifying a series of transformations that converts the model of a nonlinear diffusive process on an evolving domain to a nonlinear diffusion equation on a fixed domain, which admits known exact solutions for certain choices of diffusivity functions. We determine expressions for critical time scales and domain growth rates such that the diffusive population never reaches the domain boundaries and hence the solution remains valid.