An estimation of level sets for non local KPP equations with delay

TL;DR

This paper analyzes the asymptotic behavior of solutions to a delayed nonlocal linear parabolic equation and applies findings to estimate level sets in nonlinear KPP equations, revealing differences from classical cases.

Contribution

It introduces new estimates for level sets of delayed nonlocal KPP equations and highlights non-persistence of solutions with compact support, contrasting classical results.

Findings

Derived asymptotic estimates for solutions of the linear delayed equation.

Applied these estimates to nonlinear KPP equations with delay.

Showed solutions may not persist with compact support, unlike classical cases.

Abstract

We study the large time asymptotic behavior of the solutions of the linear parabolic equation with delay $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + \int_{\mathbb{R}} k(x-y) \, u (t-h, y)\, dy$, $x \in \R$, $\ t >0$, and $k(x) \in L^1(\R)$. As an application we get estimates on the measure of level sets of non local KPP type equations with delay. For this type of nonlinear equations we prove that, in contrast with the classical case, the solution to the initial value problem with data of compact support may not be persistent.