Stable manifold approximation for the heat equation with nonlinear boundary condition (Q1592764)

The authors consider positive solutions of the heat equation \(u_t=u_{xx}\) in \((0,1)\times(0,T)\) complemented by the Neumann boundary conditions \(u_x=0\) at \(x=0\), \(u_x=u^p-u\) at \(x=1\), \(p>1\), and the initial condition \(u(\cdot,0)=u_0\). The function \(u\equiv 1\) is the unique positive steady state and its stable manifold \(W^s\) has codimension 1. Moreover, given any \(u_0>0\) there exists a unique \(\lambda_c>0\) such that \(\lambda_cu_0\in W^s\). The authors use and analyze a semidiscrete numerical scheme in order to approximate the value of \(\lambda_c\). Numerical experiments are presented for \(p=2\) and \(u_0(x)={1\over 2}\cos(2\pi x)+1\).