Pullback Permanence for Non-Autonomous Partial Differential Equations
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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 ut - Δu = λu - b(t)u3, with b(t) > 0 for all t ∈ ℝ, limt→∞ 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.
CitationLanga, J. A., & Suarez, A. (2002). Pullback permanence for non-autonomous partial differential equations. Electronic Journal of Differential Equations, 2002(72), pp. 1-20.
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