We consider a thin-film equation modeling the epitaxial growth of nanoscale thin films. By exploiting the boundary conditions and the variational structure of the equation, we look for conditions on initial data that ensure the solution exists globally or blows up in finite time. Moreover, for global solution, we establish the exponential decays of solutions and energy functional, and give the concrete decay rate. As for blow-up solution, we prove that the solution grows exponentially and obtain the behavior of energy functional as t tends to the maximal existence time. Under the low initial energy, we get further two necessary and sufficient conditions for the solution existing globally and blowing up in finite time, respectively. A new sufficient condition such that the solution exists globally is obtained; we point out that this initial condition is independent to initial energy. Finally, we discuss the vacuum isolating phenomena of the solution.

中文翻译：

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一类薄膜方程的整体存在，有限时间爆破和真空隔离现象

我们考虑了一个模拟纳米级薄膜外延生长的薄膜方程。通过利用边界条件和方程的变分结构，我们在初始数据中寻找条件，以确保解决方案全局存在或在有限时间内爆炸。此外，对于整体解，我们建立了溶液的指数衰减和能量函数，并给出了具体的衰减率。对于爆破解决方案，我们证明了该解决方案呈指数增长，并获得了能量函数t的行为。倾向于最大生存时间。在较低的初始能量下，对于存在于全球且在有限时间内爆炸的解决方案，我们分别又获得了两个充要条件。获得一个新的充分条件，使解决方案全局存在；我们指出，这种初始条件与初始能量无关。最后，我们讨论了溶液的真空隔离现象。