New invariant sets to nonlinear diffusion equations with x-dependent convection and absorption

In this paper, we introduce a new invariant set Ω = { u ( x , t ) ∈ C ∞ | D x H ( x , u ) = 0 } ( H u ≠ 0 ) , where H is a smooth function of x and u to be determined. The invariant sets and exact solutions to nonlinear diffusion equation u t = ( A ( u ) u x ) x + Q ( x , u ) u x + P ( x , u ) , are discussed. The forms of A ( u ) , Q ( x , u ) and P ( x , u ) which leads to invariant solutions are determined. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Ω. Another new invariant set is described, it is used to some nonlinear evolution equations.