EECT
Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term
Aibo Liu Changchun Liu
Evolution Equations & Control Theory 2015, 4(3): 315-324 doi: 10.3934/eect.2015.4.315
In this paper, we consider the Cauchy problem of a sixth order Cahn-Hilliard equation with the inertial term, \begin{eqnarray*} ku_{t t} + u_t - \Delta^3 u - \Delta(-a(u) \Delta u -\frac{a'(u)}2|\nabla u|^2 + f(u))=0. \end{eqnarray*} Based on Green's function method together with energy estimates, we get the global existence and optimal decay rate of solutions.
keywords: inertial term. Sixth order Cahn-Hilliard existence optimal decay rate
EECT
Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions
Changchun Liu Hui Tang
Evolution Equations & Control Theory 2017, 6(2): 219-237 doi: 10.3934/eect.2017012

In this paper, we discuss the existence of the periodic solutions of a Cahn-Hillard/Allen-Cahn equation which is introduced as a simplification of multiple microscopic mechanisms model in cluster interface evolution. Based on the Schauder type a priori estimates, which here will be obtained by means of a modified Campanato space, we prove the existence of time-periodic solutions in two space dimensions. The uniqueness of solutions is also discussed.

keywords: Cahn-Hilliard/Allen-Cahn equation existence uniqueness periodic solution
CPAA
Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions
Changchun Liu Zhao Wang
Communications on Pure & Applied Analysis 2014, 13(3): 1087-1104 doi: 10.3934/cpaa.2014.13.1087
In this paper, we study the time periodic solution of a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Based on Leray-Schauder's fixed point theorem and Campanato spaces, we prove the existence of time-periodic solutions in two space dimensions.
keywords: time-periodic solution existence. Sixth order parabolic
CPAA
A fourth order nonlinear degenerate parabolic equation
Changchun Liu
Communications on Pure & Applied Analysis 2008, 7(3): 617-630 doi: 10.3934/cpaa.2008.7.617
In this paper, we study a fourth order degenerate parabolic equation, which arises in epitaxial growth of nanoscale thin films. We establish the existence of weak solutions, based on the uniform estimates for the approximate solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.
keywords: existence degenerate parabolic equation Fourth order finite speed of propagation.
CPAA
Existence of weak solutions for a generalized thin film equation
Changchun Liu Jingxue Yin Juan Zhou
Communications on Pure & Applied Analysis 2007, 6(2): 465-480 doi: 10.3934/cpaa.2007.6.465
In this paper, we consider an initial-boundary problem for a fourth-order nonlinear parabolic equations. The problem as a model shares the scaling properties of the thin film equation, or as a model arises in epitaxial growth of nanoscale thin films. Our approach lies in the combination of the energy techniques with some methods based on the framework of Campanato spaces. Based on the uniform estimates for the approximate solutions, we establish the existence of weak solutions.
keywords: existence Thin film equation Campanato space.

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