Lie group classification and invariant solutions of a class of quasilinear evolution equations with nonlinear gradient-power flux and state-dependent diffusivity

In this paper, we perform a Lie group classification of the quasilinear evolution equation (forumal present) where Φ(u) is a state-dependent diffusivity and F(u) is an arbitrary reaction term. The kernel Lie algebra admitted for arbitrary nonconstant Φ(u) and F(u) is identified. All symmetry extensions beyond the kernel algebra are then classified with respect to the arbitrary elements, and the corresponding infinitesimal generators are explicitly constructed. For each classified symmetry algebra, one-dimensional optimal systems of subalgebras are obtained and used to derive similarity reductions of the governing equation. Several reduced ordinary differential equations are solved explicitly, leading to exact invariant solutions for selected nonlinear models in the classification. In addition, conservation laws are constructed by the multiplier method for the admissible reaction case, yielding explicit conserved vectors.