Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method,the technique of equivalence transfor-mations,and the theory of classification of abstract low-dimensional Lie algebras.We show that there are three equations admitting simple Lie algebras of dimension three.All non-equivalent equations admitting simple Lie algebras are nothing but these three.Furthermore,we also show that there exist two,five,twenty-nine and twenty-six non-equivalent third-order nonlinear evolution equations admitting one-,two-,three-,and four-dimensional solvable Lie algebras,respectively.