  This paper addresses input-to-state stability (ISS) properties with respect
to boundary and in-domain disturbances for a class of semi-linear partial
differential equations (PDEs) subject to Dirichlet boundary conditions. The
developed approach is a combination of the method of De Giorgi iteration and
the technique of Lyapunov functionals by adequately splitting the original
problem into two subsystems. The ISS in $L^\infty$-norm for a 1-$D$ transport
equation and the ISS in $L^2$-norm for Burgers' equation have been established
using this method. As an application of the main result for the 1-D transport
equation, a study on ISS properties in $L^\infty$-norm of a 1-D transport
equation with anti-stable term under boundary feedback control is carried out.
This is the first time that the method of De Giorgi iteration is introduced in
the ISS theory for infinite dimensional systems, and the developed method can
be generalized for tackling some problems on multidimensional spatial domains
and be applied to a wider class of nonlinear parabolic PDEs.
