On nonlinear diffusion equations with x-dependent convection and absorption Original Research Article

For the 1+1-dimensional nonlinear diffusion equations with x-dependent convection and source terms ut=(D(u)ux)x+Q(x,u)ux+P(x,u), we obtain conditions under which the equations admit the second-order generalized conditional symmetries η(x,u)=uxx+H(u)ux2+G(x,u)ux+F(x,u) and the first-order sign-invariants J(x,u)=ut−A(u)ux2−B(x,u)ux−C(x,u) on the solutions u(x,t). Several different generalized conditional symmetries and first-order sign-invariants for equations in which the diffusion term offers different possibilities (power-law, exponential, Mullin, Fujita) are presented. Exact solutions to the resulting equations corresponding to the generalized conditional symmetries and the first-order sign-invariants are constructed.