Optimal systems and group classification of (1+2)-dimensional heat equation

The optimal systems and symmetry breaking interactions for the (1 + 2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H-2. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H-2. A list of representatives of all Lie subalgebras of the Lie algebra h(2) of the Lie group H-2 is given in the form of tables and many of their properties are established. We derive the most general interactions F(t, x, y, u, u(x), u(y)) such that the equation u(t) = u(xx) + u(yy) + F(t, x, y, u, u(x), u(y)) is invariant under each subgroup.