Asymptotic stability of solutions to semilinear evolution equations in Banach spaces

TL;DR

This paper establishes a new linearization principle for the nonlinear stability of solutions to semilinear parabolic evolution equations in Banach spaces, demonstrating exponential convergence to equilibria under broad conditions.

Contribution

It introduces a novel linearization approach applicable to a wide class of semilinear evolution equations with manifold equilibria, extending stability analysis beyond traditional assumptions.

Findings

Solutions close to the equilibrium manifold converge exponentially.

Weak solutions of fluid-solid interaction equations stabilize to steady states.

Convergence occurs in multiple Sobolev spaces with explicit rates.

Abstract

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 $H^{2}_2(\Omega)$.