The purpose of this presentation is to study quasi linear parabolic partial differential equations of second order, on a bounded junction, satisfying a nonlinear and non dynamical Neumann boundary condition at the junction point. We prove the existence and the uniqueness of a classical solution. This is the first result of this type in literature. We will explain why we face out with a new problem which differs from the other ones in the theory of quasilinear parabolic equations. The main idea is to build a solution with an ellipitc Scheme. Several technical methods in analysis and the theory of PDEs are used for instance: comparison theorem, Bernstein gradient's method, gradient barrier functions, and Schauder's estimates.