Time periodic solutions of a class of degenerate parabolic equations

2000 ◽  
Vol 16 (2) ◽  
pp. 180-187 ◽  
Author(s):  
Wang Yifu ◽  
Wu Zhuoqun ◽  
Yin Jingxue
Keyword(s):  
Periodic Solutions ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Time Periodic Solutions ◽  
Degenerate Parabolic ◽  
Time Periodic

Time-periodic solutions to semilinear parabolic equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 329-342 ◽  
Author(s):  
Peter Grindrod ◽  
Bryan P. Rynne
Keyword(s):  
Periodic Solutions ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Periodic Forcing ◽  
Linear Diffusion ◽  
Linear Evolution ◽  
Time Periodic Solutions ◽  
Semilinear Parabolic ◽  
Time Periodic ◽  
Linear Evolution Equations

SynopsisWe consider a class of non-linear evolution equations subject to a periodic forcing term. Using bifurcation theory we obtain results on the existence and number of periodic solutions. The theory applies to semi-linear diffusion equations defined on bounded or unbounded domains.

Coexistence Solutions for a Periodic Competition Model with Singular–Degenerate Diffusion

2016 ◽  
Vol 60 (4) ◽  
pp. 1065-1075
Author(s):  
Yifu Wang ◽  
Jingxue Yin ◽  
Yuanyuan Ke
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Parabolic Equations ◽  
Competition Model ◽  
Degenerate Parabolic Equations ◽  
Degenerate Diffusion ◽  
Volterra Type ◽  
Non Local ◽  
Degenerate Parabolic

AbstractWe investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

scholarly journals A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Jiebao Sun ◽  
Dazhi Zhang ◽  
Boying Wu
Keyword(s):  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Volterra Model ◽  
Generalized Solutions ◽  
Degenerate Parabolic Equations ◽  
Volterra System ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Time Periodic

We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.

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

2015 ◽  
Vol 17 (02) ◽  
pp. 1450025 ◽  
Author(s):  
Genni Fragnelli ◽  
Dimitri Mugnai ◽  
Paolo Nistri ◽  
Duccio Papini
Keyword(s):  
Periodic Solutions ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Degree Theory ◽  
Biological Species ◽  
Dirichlet Boundary Conditions ◽  
Degenerate Parabolic Equations ◽  
Topological Degree Theory ◽  
Degenerate Parabolic ◽  
Nonlocal Terms

We study the existence of nontrivial, nonnegative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray–Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.

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