Periodic solutions of a class of degenerate parabolic system with delays

Abstract

This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…Similar topological methods are also employed to a great extent for the existence of non-negative periodic solutions of degenerate and doubly degenerate parabolic equations, see [3], [9], [20], [30], [31], [38], [42], [44], [45], [48], [55], [56], [58], [59], [60], [61], [62], [63], [64], [67], [68]. Nonlocal models to study aggregation in biological systems with degenerate diffusion are proposed in several papers, see the recent [12], [43] and the references therein.…”

Convergence to equilibrium in degenerate parabolic equations with delay

“…There are also many works dealing with the case of linear diffusion and nonlinear delay terms (r > 0), see [19,24,29] for an overview and references. More recently researchers have considered problems incorporating degenerate diffusion, delay and periodic forcing [28,30].…”

Global Asymptotic Behavior of a Nonautonomous Competitor-Competitor-Mutualist Model

“…Zhou and Fu [13] investigated the global asymptotic behavior of the time-dependent solutions and the existence of periodic solutions for the model with discrete delays. Very recently, replacing the usual −Δ term by a degenerate elliptic operator as −Δ , Wang and Yin [14] proved the existence of maximal and minimal -periodic solutions to the model with time delays by the Schauder fixed point theorem. It is important to note that the uniqueness of positive periodic solution is not considered in the previous references.…”