Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity

• Published in: Communications in nonlinear science & numerical simulation 2016-07, Vol.36, p.98-108
• Main Authors: Cherniha, Roman, King, John R., Kovalenko, Sergii
• Format: Article
• Language: English
• Subjects: Descriptions; Diffusivity; Exact solution; Gradient-dependent diffusivity; Invariance; Lie groups; Lie symmetry; Mathematical analysis; Nonlinear diffusion equation; Nonlinearity; Reaction-diffusion equations; Symmetry; The Perona–Malik equation
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2015.11.023

•A complete Lie symmetry classification of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity is obtained.•A subclass of these equations, admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable, is uncovered.•Symmetries and reductions of a diffusion equation in 2-D space, often called the Perona-Malik equation, are studied in detail.•The Lie symmetries are applied in a search for exact solutions of the equation with power-law diffusivity. Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on the form of diffusivity and source (sink) in the equations is derived. It is established that there exists a subclass in 1-D space admitting an infinite-dimensional Lie algebra of invariance so that it is linearisable. A special power-law diffusivity with a fixed exponent, which leads to wider Lie invariance of the equations in question in 2-D space, is also derived. However, it is shown that the diffusion equation without a source term (which often arises in applications and is sometimes called the Perona–Malik equation) possesses no rich variety of Lie symmetries depending on the form of gradient-dependent diffusivity. The results of the Lie symmetry classification for the reduction to lower dimensionality, and a search for exact solutions of the nonlinear 2-D equation with power-law diffusivity, also are included.