Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set ?E₀={u:u_x=f^ˊ(x)F(u)+ε g^ˊ(x) -f^ˊ(x)g(x)]F(u)exp(-∫^u(1/F(z))dz), where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact solutions to nonlinear diffusion equation u_t=(D(u)u_x)_x+Q(x,u)u_x+P(x,u), are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set ?E₀.