A Simple Unified Approach to Some Convergence Theorems of L. Simon

We give a new and easier proof for the convergences result of L. Simon [15] concerning global and bounded solutions of a heat equation with analytic nonlinearity. By constructing a new Liapunov functional for the elliptic equation, we are able to prove the second convergence theorem of [15] in the same way as the first theorem. The results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type.