  In a series of recent works by Demirkaya et al. stability analysis for the
static kink solutions to the 1D continuous and discrete Klein-Gordon equations
with a $\mathcal{PT}$-symmetric perturbation has been analysed. We consider the
linear stability problem for the static kink in 2D Klein-Gordon field taking
into account spatially localized $\mathcal{PT}$-symmetric perturbation. The
perturbation is in the form of viscous friction, which does not affect the
static solutions to the unperturbed Klein-Gordon equation. Small dynamic
perturbation around the static kink solution is considered to formulate the
linear stability problem. The effect of the small perturbation on the solutions
to the corresponding eigenvalue problem is analysed. The main result is
presented in the form of a theorem describing the behavior of the eigenvalues
corresponding to the extended and localised eigenmodes as the functions of the
perturbation parameter.
