  This paper addresses the input-to-state stability (ISS) and integral
input-to-state stability (iISS) for a class of nonlinear higher dimensional
parabolic partial differential equations (PDEs) with different types of
boundary disturbances (Robin or Neumann or Dirichlet) from different spaces by
means of approximations of Lyapunov functions. Specifically, by constructing
approximations of (coercive and non-coercive) ISS Lyapunov functions we
establish: (i) the ISS and iISS in $L^1$-norm (and weighted $L^1$-norm) for
PDEs with boundary disturbances from
$L^q_{loc}(\mathbb{R}_+;L^1(\partial\Omega))$-space for any $q\in [1,+\infty]$;
(ii) the iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary
disturbances from $L^\Phi_{loc}(\mathbb{R}_+;L^1(\partial\Omega))$-space for
certain Young function $\Phi$; and (iii) the ISS and iISS in $L^\Phi$-norm (and
weighted $K_\Phi$-class) for PDEs with boundary disturbances from
$L^q_{loc}(\mathbb{R}_+;K_\Phi(\partial\Omega))$-class for any $q\in
[1,+\infty]$ and certain Young function $\Phi$. The ISS properties stated in
(ii) and (iii) are assessed in the framework of Orlicz space or Orlicz class.
