薄膜型方程的定态能级
Energy levels of steady states for thin film type equations
摘要：
本文利用耗散能（Liapunov函数）研究了发展方程h_t =-（f（h）h_{xxx}）_x-（g（h）h_x）_x的相空间.这里h（x，t）是非负的，在h = 0时，系数函数f> 0和g可以退化为0，或者爆炸为无穷大，或者趋于非零常数。我们首先证明了所有的正周期稳定状态是“能量不稳定”的不动点的演变（这意味着一些零平均扰动下的能量减少），如果g/f是凸的，或者如果扰动被允许有周期长于稳定状态。对于幂律系数（f（y）= y ^n和g（y）= B y ^m，对于某些B> 0），我们解析地确定了不同稳态的相对能级。例如，对于[1，2]中的m-n，适当选择周期和平均值，我们发现三个根本不同的稳态。第一个是一个恒定的稳态，是非线性稳定的，是一个局部最小的能量。第二个是一个正的周期稳定状态，是线性不稳定的，并具有更高的能量比恒定的稳定状态;它是一个鞍点。第三种是周期性收集的“液滴”（compounds支持）的稳定状态具有较低的能量比正的稳定状态或恒定的。由于能量必须沿每个轨道减少，这些结果显着约束的演化方程的动力学。我们的研究结果表明，异宿连接可能存在于某些稳定状态之间，例如从周期稳定状态到液滴稳定状态。在一篇配套文章中，我们进行了数值模拟，以确认它们的存在。
英文摘要：
We study the phase space of the evolution equation h_t = -(f(h) h_{xxx})_x - (g(h) h_x)_x by means of a dissipated energy (a Liapunov function). Here h(x,t) is nonnegative, and at h=0 the coefficient functions f>0 and g can either degenerate to 0, or blow up to infinity, or tend to a nonzero constant. We first show all positive periodic steady states are 'energy unstable' fixed points for the evolution (meaning the energy decreases under some zero--mean perturbation) if g/f is convex or if the perturbations are allowed to have period longer than that of the steady state. For power law coefficients (f(y) = y^n and g(y) = B y^m for some B > 0) we analytically determine the relative energy levels of distinct steady states. For example, with m-n in [1,2) and for suitable choices of the period and mean value, we find three fundamentally different steady states. The first is a constant steady state that is nonlinearly stable and is a local minimum of the energy. The second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state; it is a saddle point. The third is a periodic collection of 'droplet' (compactly supported) steady states having lower energy than either the positive steady state or the constant one. Since the energy must decrease along every orbit, these results significantly constrain the dynamics of the evolution equation. Our results suggest that heteroclinic connections could exist between certain of the steady states, for example from the periodic steady state to the droplet one. In a companion article we perform numerical simulations to confirm their existence.
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