We consider a relaxation of the viscous Cahn-Hilliard equation induced by thesecond-order inertial term~$u_{tt}$. The equation also contains a semilinearterm $f(u)$ of "singular" type. Namely, the function $f$ is defined only on abounded interval of ${\mathbb R}$ corresponding to the physically admissiblevalues of the unknown $u$, and diverges as $u$ approaches the extrema of thatinterval. In view of its interaction with the inertial term $u_{tt}$, the term$f(u)$ is difficult to be treated mathematically. Based on an approachoriginally devised for the strongly damped wave equation, we propose a suitableconcept of weak solution based on duality methods and prove an existenceresult.
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机译：我们考虑由二阶惯性项〜$ u_ {tt} $引起的粘性Cahn-Hilliard方程的松弛。该方程还包含“奇异”类型的半线性项$ f（u）$。即，仅在对应于未知$ u $的物理允许值的$ {\ mathbb R} $的一定间隔上定义函数$ f $，并且在$ u $接近该间隔的极值时发散。鉴于其与惯性项$ u_ {tt} $的相互作用，因此很难对项$ f（u）$进行数学处理。在最初针对强阻尼波动方程设计的一种方法的基础上，我们提出了一种基于对偶方法的弱解的合适概念，并证明了存在性。
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