scholarly journals Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion

2004 ◽  
Vol 10 (3) ◽  
pp. 719-730 ◽  
Author(s):  
Y. S. Choi ◽  
◽  
Roger Lui ◽  
Yoshio Yamada ◽  
◽  
...  
Keyword(s):  
Global Solutions ◽  
Strongly Coupled ◽  
Cross Diffusion

Unique continuation for a reaction-diffusion system with cross diffusion

2019 ◽  
Vol 27 (4) ◽  
pp. 511-525 ◽  
Author(s):  
Bin Wu ◽  
Ying Gao ◽  
Zewen Wang ◽  
Qun Chen
Keyword(s):  
Unique Continuation ◽  
Reaction Diffusion ◽  
Cauchy Data ◽  
Carleman Estimate ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Lateral Boundary ◽  
Strongly Coupled ◽  
Cross Diffusion ◽  
Coupled Reaction

Abstract This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

scholarly journals Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers

2012 ◽  
Vol 2012 ◽  
pp. 1-25
Author(s):  
Yujuan Jiao ◽  
Shengmao Fu
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Global Solutions ◽  
Energy Estimates ◽  
Positive Equilibrium ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

scholarly journals Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density

2021 ◽  
Vol 2131 (3) ◽  
pp. 032043
Author(s):  
M Aripov ◽  
A S Matyakubov ◽  
J O Khasanov ◽  
M M Bobokandov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Variable Density ◽  
Divergence Form ◽  
Cross Diffusion ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

Abstract In this paper the properties of solutions of nonlinear parabolic equation not in divergence form | x | − 1 ∂ u ∂ t = u q ∂ ∂ x ( | x | n u m − 1 | ∂ u k ∂ x | p − 2 ∂ u ∂ x ) + | x | − 1 u β are studied. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic equation not in divergence form, depending on the value in the equation of the numerical parameters necessary and sufficient signs of their existence. The compactly supported solution of the Cauchy problem for a cross-diffusion parabolic equation not in divergence form with a source and a variable density is obtained.

EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY–PREDATOR MODEL WITH CROSS-DIFFUSION

2010 ◽  
Vol 03 (02) ◽  
pp. 161-172 ◽  
Author(s):  
SHENGHU XU ◽  
WEIDONG LV
Keyword(s):  
Neumann Boundary Condition ◽  
Space Dimension ◽  
Existence Of Solutions ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Energy Estimates ◽  
Cross Diffusion ◽  
Global Existence Of Solutions ◽  
Homogeneous Neumann Boundary Condition ◽  
Predator Model

In this paper, a ratio-dependent prey–predator model with cross-diffusion and homogeneous Neumann boundary condition is studied. Using the energy estimates and the bootstrap arguments, the global existence of solutions for the model is investigated when the space dimension is less than ten.

TURING PATTERNS OF A PREDATOR–PREY MODEL WITH A MODIFIED LESLIE–GOWER TERM AND CROSS-DIFFUSIONS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250060 ◽  
Author(s):  
GUANG-PING HU ◽  
XIAO-LING LI
Keyword(s):  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Steady States

In this paper, a strongly coupled diffusive predator–prey system with a modified Leslie–Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

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