We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

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具有非守恒质量的薄膜型方程的稳态和动力学

我们研究了一维质量不守恒的薄膜型方程的稳态和动力学。演化方程是非线性四阶简并抛物线偏微分方程 (PDE)，由允许冷凝或蒸发的挥发性粘性流体膜模型驱动。我们表明，通过改变非守恒通量的符号并打破梯度流动结构，该问题可以表现出新颖的行为，包括具有两种不同类别的共存稳态解。对这些稳态的分岔结构及其稳定性的详细分析揭示了动力学的几种可能性。对于某些参数方案，解可能导致有限时间破裂奇点。有趣的是，