  This paper is devoted to two issues. One is to provide Lyapunov-based tools
to establish integral input-to-state stability (iISS) and input-to-state
stability (ISS) for some classes of nonlinear parabolic equations. The other is
to provide a stability criterion for interconnections of iISS parabolic
systems. The results addressing the former problem allow us to overcome
obstacles arising in tackling the latter one. The results for the latter
problem are a small-gain condition and a formula of Lyapunov functions which
can be constructed for interconnections whenever the small-gain condition
holds. It is demonstrated that for interconnections of partial differential
equations, the choice of a right state and input spaces is crucial, in
particular for iISS subsystems which are not ISS. As illustrative examples,
stability of two highly nonlinear reaction-diffusion systems is established by
the the proposed small-gain criterion.
