Symmetry analysis and group-invariant solutions to inhomogeneous nonlinear diffusion equation

• Point symmetries for inhomogeneous nonlinear diffusion equation is classified.
• Due to the symmetries, optimal systems of one-dimensional subalgebra for this equation are presented.
• Explicit group-invariant solutions are derived.
• The properties related to blow-up, extinction and asymptotic behavior for these solutions are also discussed.

A classification of point symmetries for inhomogeneous nonlinear diffusion equation is discussed. The optimal systems of one-dimensional subalgebra for the equation are constructed. Explicit group-invariant solutions are derived by corresponding symmetry reductions. These solutions include static solutions, separable solutions and functionally separable solutions. The behaviors of blow-up, extinction and asymptotical behavior for these solutions are also described.