Asymptotic behavior for a heat conduction problem with perfect-contact boundary condition

In this paper we consider the heat conduction problem for a slab represented by the interval $[0,1]$. The initial temperature is a positive constant, the flux at the left end is also a positive constant, and at the right end there is a perfect contact condition: $u_{x}(1,t)+gamma u_{t}(1,t)=0$. We analyze the asymptotic behavior of these problems as $gamma$ approaches infinity, and present some numerical calculations