Conservations laws for a porous medium equation through nonclassical generators

• The classes of nonlinearly self-adjoint porous medium equations are obtained.
• Conservation laws for some of these equations, associated to classical and nonclassical generators, are established.

In Ibragimov (2007) [13] a general theorem on conservation laws was proved. In Gandarias (2011) and Ibragimov (2011) [7] and [15] the concepts of self-adjoint and quasi self-adjoint equations were generalized and the definitions of weak self-adjoint equations and nonlinearly self-adjoint equations were introduced. In this paper, we find the subclasses of nonlinearly self-adjoint porous medium equations. By using the property of nonlinear self-adjointness, we construct some conservation laws associated with classical and nonclassical generators of the differential equation.