Pullback permanence for non-autonomous partial differential equations

Langa, Jose A. ; Suárez, Antonio

Electronic journal of differential equations, Tome 2002 (2002)

Résumé

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 $t\in \mathbb{R}, \lim_{t\to \infty }b(t)=0$. 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.

Détail

Classification : 35B05, 35B22, 35B41, 37L05
Keywords: non-autonomous differential equations, pullback attractors, comparison techniques, permanence