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Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations. (English) Zbl 1171.34037
Consider the so-called higher-order convective Cahn-Hilliard equation $$ u_t - \nu uu_x + (Q(u)+\varepsilon^2u_{xx})_{xxxx}=0 $$ together with the standard Cahn-Hilliard equation $$ u_t + (Q(u)+\varepsilon^2u_{xx})_{xx}=0.$$ 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 $$ \lim_{x\to\pm\infty} = \mp\sqrt{A} $$ with $A$ some integration constant. The whole paper is concerned only with stationary solutions in one dimension hence with various asymptotics with respect to $\varepsilon$ and $\nu$.

34E15Asymptotic singular perturbations, general theory (ODE)
34B15Nonlinear boundary value problems for ODE
34E05Asymptotic expansions (ODE)
65P99Numerical problems in dynamical systems
34B40Boundary value problems for ODE on infinite intervals
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