Existence and stability of positive solutions in a parabolic problem with a nonlinear incoming flux on the boundary

Abstract

In this paper, we consider a parabolic problem with a nonlinear boundary condition which is induced by the incoming flux on the boundary. We focus on analyzing the existence and stability of bifurcating positive solutions emanating from trivial solutions. Our approach combines the Lyapunov-Schmidt method with classical local bifurcation theory, extending the framework established by Crandall and Rabinowitz. The results provide new insights into the structure and stability properties of solutions under nonlinear flux boundary effects.