Global Stability for a Class of Nonlinear PDE with Non-local Term

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.

On Global Asymptotic Stability for the LSW Model with Subcritical Initial Data

“…We prove a local existence and uniqueness theorem for the initial value problem for (1.23), where the function t → ρ(ξ(·, t)) in (1.23) is replaced by the function t → ρ(ξ(·, t), η(t)), with ρ(ζ(·), η) given by (2.20). We shall follow the same line of argument as in Lemma 2.1 of [3]. Thus following (1.9) of [3], we define for m = 1, 2, .…”

“…We proceed in parallel to the argument followed in §3 of [3]. We first linearize (3.1) with ρ(ζ(·), η) given by (2.20), (2.25) about the equilibrium ξ p (·) and study its stability.…”

“…We proceed now as in [3] to obtain properties of the function h(·), which will imply that the solution to (4.7) satisfies lim t→∞ u(t) = 0.…”

“…We accomplish this by establishing asymptotic stability results for a class of PDE similar to the one studied in [3]. In particular, we consider for some ε 0 > 0 the evolution PDE (1.23) ∂ξ(y, t) ∂t − h(y) − h(y) ∂ξ(y, t) ∂y + ρ(ξ(·, t)) ξ(y, t) − y ∂ξ(y, t) ∂y = 0, y > ε 0 , t > 0, with given non-negative initial data ξ(y, 0), y > ε 0 .…”

“…In particular, we consider for some ε 0 > 0 the evolution PDE (1.23) ∂ξ(y, t) ∂t − h(y) − h(y) ∂ξ(y, t) ∂y + ρ(ξ(·, t)) ξ(y, t) − y ∂ξ(y, t) ∂y = 0, y > ε 0 , t > 0, with given non-negative initial data ξ(y, 0), y > ε 0 . We assume as in [3] that the function h : [ε 0 , ∞) → R has the properties:…”