Global Stability for a Class of Nonlinear PDE with non-local term

TL;DR

This paper proves global asymptotic stability for a class of nonlinear PDEs similar to the Lifschitz-Slyozov-Wagner model by linking stability to a differential delay equation and using optimal control methods.

Contribution

It introduces a novel approach to stability analysis of nonlinear PDEs without Lyapunov functions, connecting PDE stability to delay equations and employing optimal control techniques.

Findings

Stability of the PDE is equivalent to stability of an associated delay equation.

Stability is established using maximal properties derived from optimal control theory.

The method avoids traditional Lyapunov function techniques.

Abstract

This paper is concerned with establishing global asymptotic stability results for a class of non-linear PDE which have some similarity to the PDE of the Lifschitz-Slyozov-Wagner model. The method of proof does not involve a Lyapounov function. It is shown that stability for the PDE is equivalent to stability for a differential delay equation. Stability for the delay equation is proven by exploiting certain maximal properties. These are established by using the methods of optimal control theory.