Vectorial quasilinear diffusion equation with dynamic boundary condition

Abstract

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)$_\varepsilon$ with a nonnegative constant $\varepsilon$, and for any $\varepsilon\ge0$, (S)$_\varepsilon$ 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 enable us to handle the transmission systems of various vectorial mathematical models, such as the Bingham type flow equations, the Ginzburg--Landau type equations, and so on. On this basis, we set the goal of this paper to prove two Main Theorems, concerned with the well-posedness of (S)$_\varepsilon$ with the precise representation of solution, and $\varepsilon$-dependence of (S)$_\varepsilon$, for $\varepsilon \ge0$.