Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 72, pp. 1-20.
Title: Pullback permanence for non-autonomous
partial differential equations
Authors: Jose A. Langa (Univ. de Sevilla, Spain)
Antonio Suarez (Univ. de Sevilla, Spain)
Abstract:
A system of differential equations is permanent if there
exists a fixed bounded set of positive states strictly
bounded away from zero to which, from a time on, any positive
initial data enter and remain. However, this fact does not happen
for a differential equation with general non-autonomous terms.
In this work we introduce the concept of pullback permanence,
defined as the existence of a time dependent set of positive
states to which all solutions enter and remain for suitable
initial time. We show this behaviour in the non-autonomous logistic
equation
$u_{t}-\Delta u=\lambda u-b(t)u^{3}$, with $b(t)>0$
for all
$t\in \mathbb{R}$, $\lim_{t\to \infty }b(t)=0$.
Moreover, a bifurcation scenario for the asymptotic behaviour of
the equation is described in a neighbourhood of the first eigenvalue
of the Laplacian. We claim that pullback permanence can be a suitable
tool for the study of the asymptotic dynamics for general
non-autonomous partial differential equations.
Submitted May 14, 2001. Published August 8, 2002.
Math Subject Classifications: 35B05, 35B22, 35B41, 37L05.
Key Words:
Non-autonomous differential equations;
pullback attractors; comparison techniques; permanence.