  We study local conservation laws of variable coefficient diffusion-convection
equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of
our investigation is the notion of equivalence of conservation laws with
respect to the equivalence groups. That is why, for the class under
consideration we first construct the usual equivalence group $G^{\sim}$ and the
extended one $\hat G^{\sim}$ including transformations which are nonlocal with
respect to arbitrary elements. The extended equivalence group $\hat G^{\sim}$
has interesting structure since it contains a non-trivial subgroup of gauge
equivalence transformations. Then, using the most direct method, we carry out
two classifications of local conservation laws up to equivalence relations
generated by $G^{\sim}$ and $\hat G^{\sim}$, respectively. Equivalence with
respect to $\hat G^{\sim}$ plays the major role for simple and clear
formulation of the final results.
