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Zbl 1171.34037
Korzec, M.D.; Evans, P.L.; Münch, A.; Wagner, B.
Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations.
(English)
[J] SIAM J. Appl. Math. 69, No. 2, 348-374 (2008). ISSN 0036-1399; ISSN 1095-712X/e

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 $\nu$.
MSC 2000:
*34E15 Asymptotic singular perturbations, general theory (ODE)
34B15 Nonlinear boundary value problems of ODE
34E05 Asymptotic expansions (ODE)
65P99 None of the above, but in this section
34B40 Boundary value problems on infinite intervals

Keywords: Cahn-Hilliard equation; stationary solution; asymptotics

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