Large time behavior and Lyapunov functionals for a nonlocal differential equation

Lyapunov Functionals for the Enskog Equation

Two Lyapunov functionals are presented for the Enskog equation. One is to describe interactions between particles with various velocities and another is to measure the L distance between two classical solutions. The former yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the L stability of glob...

Large Time Behavior for a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

Abstract. We study the large time behavior of nonnegative solutions of the Cauchy problem ut = ∫ J(x − y)(u(y, t) − u(x, t)) dy − u, u(x, 0) = u0(x) ∈ L , where |x|u0(x) → A > 0 as |x| → ∞. One of our main goals is the study of the critical case p = 1 + 2/α for 0 < α < N , left open in previous articles, for which we prove that t|u(x, t) − U(x, t)| → 0 where U is the solution of the heat equati...

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

Construction of Lyapunov Functionals for Networks of Coupled Delay Differential and Continuous-Time Difference Equations

To address various types of delays including the neutral-type arising in dynamical networks, this paper deals with coupled delay differential and continuous-time difference equations and develops stability and robustness criteria. Subsystems described by differential equations are not required to be input-to-state stable. No assumptions on network topology are made. To tackle networks in such a...

Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: The subcritical case

This is a joint work with A. Salort and N. Wolanski. We continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption { ut = Lu− u in R × (0,∞), u(x, 0) = u0(x) in R , where p > 1, u0 ≥ 0 and bounded and Lu(x, t) = ∫ J(x− y) (u(y, t)− u(x, t)) dy with J ∈ C∞ 0 (R ), radially symmetric, J ≥ 0 with ∫ J = 1. Our assumption on the initial d...