In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)${}_{\epsilon }$ with a nonnegative constant $\epsilon$, and for any $\epsilon \ge 0$, (S)${}_{\epsilon }$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma :=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method, which can...