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Asymptotic behavior near transition fronts for equations of generalized Cahn-Hilliard form. (English) Zbl 1117.35009

The paper considers a class of one-dimensional Cahn-Hilliard (CH) equations (well-known dynamical models of the spinodal decay in metallic alloys and similar physical media), such as \[ u_t=\frac{\partial}{\partial x} \left[\left(\frac 32 u^2-\frac 12\right)u_x\right]-u_{xxxx}, \]
which admit a solution in the form of a kink. In the case of the equation written above, the kink is \(u_{\text{kink}}=\tanh(x/2)\). It is known that kink solutions of equations of CH type are stable. This work aims to establish more accurate results for the asymptotic stability of the kink, using the method of Evans function, realized in terms of the analysis of perturbations around the kink by means of Green’s function. The main result reported in the paper is a rigorous proof of the fact that, if the initial perturbation added to the kink decays at \(| x| \to\infty\) as \((1+| x| )^{3/2}\) or faster, the solution (possibly, with a shifted center) will be different from the unperturbed kink by a perturbation decaying as \((1 + | x| + \sqrt{t})^{-3/2}\) or faster.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K30 Initial value problems for higher-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
35K55 Nonlinear parabolic equations
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