Help with an article on convex-splitting method

Question

I trying to simulate gradient flow $$\frac{\partial u}{\partial t}=-\nabla_x F(u)$$ where F(u) is (in my case): $$F(u)=2u^4-u^2$$ for u $\in [-0.5,0.5]$. Since the equation is non-linear, I would like to use convex-splitting method where the potential F(u) is split into two convex potentials: $$F(u)=F_c -F_e,$$ and then the inital equation can be written as: $$\frac{u^{n+1}-u^{n}}{\Delta t}= \nabla F_e(u^{n})-\nabla F_c(u^{n+1})$$ I found this article: Eyre, David J.. “An Unconditionally Stable One-Step Scheme for Gradient Systems.” IEEE Transactions on Image Processing (1997): n. pag. and based on the method described on pages 8-9 I calculated matrix A to be: $$A=J(\nabla F(u))(u) = 24 u^2-2.$$ F(u) has minima (minimums?) at +/- 0.5 which gives me: $$F_c=\frac{1}{2}<Au,u> = 2u^2,$$ and $$F_e=F_c-F=-2u^4+3u^2.$$ The equation can then be written as: $$\frac{u^{n+1}-u^{n}}{\Delta t}= -4u^{n+1}-8(u^n)^3+6(u^n)$$ This however diverges when I try to simulate it (finite element method). If I use $F(u_0)=0$ where $u_0=+/- \frac{1}{\sqrt 2}$ I get: $A=10$ and: $$F_c=\frac{1}{2}<Au,u> = 5u^2,$$ and $$F_e=F_c-F=-2u^4+6u^2$$ and this diverges too. What am I doing wrong?