  Recently, the problem of boundary stabilization for unstable linear
constant-coefficient reaction-diffusion equation on N-balls has been solved by
means of the backstepping method. However, the extension of this result to
spatially-varying coefficients is far from trivial. This work deals with
radially-varying reaction coefficients under revolution symmetry conditions on
a disk (the 2-D case). Under these conditions, the equations become singular in
the radius. When applying the backstepping method, the same type of singularity
appears in the backstepping kernel equations. Traditionally, well-posedness of
the kernel equations is proved by transforming them into integral equations and
then applying the method of successive approximations. In this case, the
resulting integral equation is singular. A successive approximation series can
still be formulated, however its convergence is challenging to show due to the
singularities. The problem is solved by a rather non-standard proof that uses
the properties of the Catalan numbers, a well-known sequence frequently used in
combinatorial mathematics.
