标题： Banach空间中半线性发展方程解的渐近稳定性 显示英文标题

标题： Asymptotic stability of solutions to semilinear evolution equations in Banach spaces

摘要： 我们证明了关于抛物型半线性演化方程解的非线性稳定性的一个新线性化原理。 我们假设平衡点集形成一个正常稳定和正常双曲平衡点的有限维流形。 此外，我们假设线性化算子是解析半群的生成元（不一定是稳定的）。 我们证明，如果我们的演化方程存在全局时间的弱解，并且在所有时间都“接近”平衡点流形，则该解最终将以指数速率收敛到一个平衡点。 我们将抽象结果应用于描述充满流体的重固体运动的方程。 在对物理配置和初始条件的一般假设下，我们证明控制方程的弱解最终以指数速率收敛到稳态。 特别是，相对于固体的流体速度在每个$p\in [1,\infty)$时在$H^{2\alpha}_p(\Omega)$中趋于$t\to\infty$，以及在$H^{2}_2(\Omega)$中也趋于$\alpha\in [0,1)$。 显示英文摘要

摘要： We prove a new linearization principle for the nonlinear stability of solutions to semilinear evolution equations of parabolic type. We assume that the set of equilibria forms a finite dimensional manifold of normally stable and normally hyperbolic equilibria. In addition, we assume that the linearized operator is the generator of an analytic semigroup (not necessarily stable). We show that if a mild solution to our evolution equation exists globally in time and remains ``close'' to the manifold of equilibria at all times, then the solution must eventually converge to an equilibrium point at an exponential rate. We apply our abstract results to the equations governing the motion of a fluid-filled heavy solid. Under general assumptions on the physical configuration and initial conditions, we show that weak solutions to the governing equations eventually converge to a steady state with an exponential rate. In particular, the fluid velocity relative to the solid converges to zero as $t\to\infty$ in $H^{2\alpha}_p(\Omega)$ for each $p\in [1,\infty)$ and $\alpha\in [0,1)$ as well as in $H^{2}_2(\Omega)$.