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.