Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations.

*(English)*Zbl 1171.34037Consider the so-called higher-order convective Cahn-Hilliard equation

$${u}_{t}-\nu u{u}_{x}+{(Q\left(u\right)+{\epsilon}^{2}{u}_{xx})}_{xxxx}=0$$

together with the standard Cahn-Hilliard equation

$${u}_{t}+{(Q\left(u\right)+{\epsilon}^{2}{u}_{xx})}_{xx}=0\xb7$$

The stationary solutions obtained by solving the resulting, by letting ${u}_{t}=0$, ordinary differential equation together with their stability are considered. They are discussed with the far-field conditions as boundary value conditions

$$\underset{x\to \pm \infty}{lim}=\mp \sqrt{A}$$

with $A$ some integration constant. The whole paper is concerned only with stationary solutions in one dimension hence with various asymptotics with respect to $\epsilon $ and $\nu $.

Reviewer: Vladimir Răsvan (Craiova)