[Paper Review] On normal stability for nonlinear parabolic equations

Jan Pruess, Gieri Simonett+1 more|arXiv (Cornell University)|Nov 11, 2008

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic equations when the set of equilibria forms a $C^1$-smooth, normally stable manifold. Using maximal regularity and a direct approach avoiding center manifold theory, it proves global existence and exponential convergence to some equilibrium on the manifold for initial data near a normally stable equilibrium.

ABSTRACT

We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.

Motivation & Objective

• To establish convergence of solutions to equilibria in quasilinear and fully nonlinear parabolic equations when the equilibrium set is a non-discrete $C^1$-manifold.
• To provide a direct, simplified proof of the generalized principle of linearized stability without relying on the technical machinery of center manifold theory.
• To extend the $L_p$-maximal regularity approach from [26] to a broader class of fully nonlinear parabolic equations.
• To demonstrate that the conditions of semi-simple zero eigenvalue and stable spectral part are sufficient for convergence to the equilibrium manifold.
• To show that the convergence is exponential and uniform for initial data in a neighborhood of the equilibrium manifold.

Proposed method

• Transform the original equation into a deviation variable $v = u - u_*$ around a given equilibrium $u_*$, leading to a perturbed equation $\dot{v} + Av = G(v)$.
• Use maximal regularity theory in Banach space settings $\mathbb{E}_0(J)$ and $\mathbb{E}_1(J)$ to control solution norms over time intervals.
• Decompose the state space into center and stable subspaces using spectral projections, with $P_c$ and $P_s$ corresponding to the kernel and stable spectral part of the linearization $A$.
• Define a contraction mapping on a suitable function space using exponential weights $e^{\sigma t}$ to handle the long-time behavior and ensure decay estimates.
• Apply non-expansive properties of the nonlinear map $\phi$ and use estimates on the nonlinear remainder $R(x,y)$ to close the contraction argument.
• Establish exponential decay of the stable component $y(t)$ and convergence of the center component $x(t)$ to a limit $x_\infty$, ensuring $v(t) \to v_\infty$ in the $X_\gamma$-norm.

Experimental results

Research questions

• RQ1 Under what conditions does a solution of a fully nonlinear parabolic equation converge to an equilibrium on a non-discrete $C^1$-manifold of equilibria?
• RQ2 Can the convergence to equilibria be proven without invoking the full center manifold theory?
• RQ3 What role does maximal regularity play in simplifying the proof of convergence in normally stable settings?
• RQ4 Are the conditions of semi-simple zero eigenvalue and stable spectral part necessary for convergence to the equilibrium manifold?
• RQ5 How does the exponential decay rate of the solution depend on the initial data and the spectral properties of the linearized operator?

Key findings

• Solutions to quasilinear and fully nonlinear parabolic equations converge exponentially to some equilibrium in the normally stable manifold for initial data sufficiently close to the manifold.
• The convergence rate is exponential, with $|u(t) - u_\infty|_\gamma \leq M e^{-\omega t} |P_s v_0 - \phi(P_c v_0)|_\gamma$ for some $\omega > 0$ and $M > 0$.
• The proof avoids center manifold theory by using maximal regularity and a direct contraction argument in weighted function spaces.
• The equilibrium manifold $\mathcal{E}$ is locally unique near $u_*$, and no other equilibria exist in a neighborhood of $u_*$.
• The conditions of semi-simple zero eigenvalue and stable spectral part of the linearization are necessary for convergence to the manifold, as shown by counterexamples in [26].
• The method applies to a broad class of equations, including geometric evolution equations, phase transitions, and free boundary problems, with uniform convergence results.