Task 1: A first closer look at the Cahn-Hilliard equation

Before we start developing a numerical method for the Cahn-Hilliard equation, let us discuss some basic properties of the so physical meaning of the different terms in the equation.

1. Plot the logarithmic potential \(F(u)\) for \(u\in (-1,1)\) for some \(\theta\) values in \([0.7, 1.6]\) with with \(\theta < \theta_c\), \(\theta = \theta_c\) and \(\theta > \theta_c\) for a critical temperature \(\theta_c = 1.5\) What do you observe?

2. Show that the Cahn-Hilliard equation is invariant under the transformation \(u \mapsto -u\), i.e. if \(u(\boldsymbol{x},t)\) is a solution, then \(-u(\boldsymbol{x},t)\) is also a solution.

3. Show that solutions of the Cahn-Hilliard equation which are periodic on a rectangular domain \(\Omega = [0,L_x)\times[0,L_y)\) are mass conservative in the sense

   \[ \dfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega} u(\boldsymbol{x},t) \;\mathrm{d}\boldsymbol{x} = 0. \]
   In other words, the total mass \(\int_{\Omega} u(\boldsymbol{x},t) \;\mathrm{d}\boldsymbol{x}\) does not change over time.

Hint: Multiply the Cahn-Hilliard equation by constant \(1\) and integrate over a rectangular domain and use the divergence theorem to perform integration by parts. What can you say about the boundary integral?