Convergence to equilibrium in degenerate parabolic equations with delay

Laister, R.; Laister, Robert

Abstract

© 2012 Elsevier Ltd In [11], Busenberg & Huang (1996) showed that small positive equilibria can undergo supercritical Hopf bifurcation in a delay-logistic reaction–diffusion equation with Dirichlet boundary conditions. Consequently, stable spatially inhomogeneous time-periodic solutions exist. Previously in [12] Badii, Diaz & Tesei (1987) considered a similar logistic-type delay-diffusion equation, but differing in two important respects: firstly by the inclusion of nonlinear degenerate diffusion of so-called porous medium type, and secondly by the inclusion of an additional ‘dominating instantaneous negative feedback’ (where terms local in time majorize the delay terms, in some sense). Sufficient conditions were given ensuring convergence of non-negative solutions to a unique positive equilibrium. A natural question to ask, and one which motivated the present work, is: can one still ensure convergence to equilibrium in delay-logistic diffusion equations in the presence of nonlinear degenerate diffusion, but in the absence of dominating instantaneous negative feedback? The present paper considers this question and provides sufficient conditions to answer in the affirmative. In fact the results are much stronger, establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay-logistic terms to larger classes of nonlinearities. Furthermore the results obtained are independent of the size of the delay.

• Journal Article Type: Article
• Publication Date: Apr 1, 2013
• Publicly Available Date: Jun 7, 2019
• Journal: Nonlinear Analysis, Theory, Methods and Applications
• Print ISSN: 0362-546X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 81
• Pages: 200-210
• DOI: https://doi.org/10.1016/j.na.2012.10.019
• Keywords: delay, degenerate, parabolic, diffusion, logistic, convergence
• Public URL: https://uwe-repository.worktribe.com/output/933228
• Publisher URL: http://dx.doi.org/10.1016/j.na.2012.10.019