Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u t = [u m (u x ) n ] x + P(u)u x + Q(u), where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η = u xx + H (u)u 2x + G(u)(u x ) 2−n + F(u)u 1−n x and the Hamilton-Jacobi sign-invariant J = u t + A(u)u n+1x + B(u)u x + C(u) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.

The conditional Lie-Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie-Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite-dimensional dynamical systems.

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