scholarly journals Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

2021 ◽  
pp. 3052-3060
Author(s):  
Maan A. Rasheed
Keyword(s):  
Dirichlet Boundary ◽  
Blow Up ◽  
Dimensional Space ◽  
Single Point ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Coupled Reaction

    This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals Reaction-advection-diffusion competition models under lethal boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kwangjoong Kim ◽  
Wonhyung Choi ◽  
Inkyung Ahn
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Reaction Diffusion ◽  
Steady States ◽  
Dirichlet Boundary Conditions ◽  
Hostile Environment ◽  
Directed Movement ◽  
Constant Rate ◽  
Homogeneous Dirichlet Boundary Conditions

<p style='text-indent:20px;'>In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.</p>

scholarly journals Blow-Up Rate Estimates for a System of Reaction-Diffusion Equations with Gradient Terms

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Maan A. Rasheed ◽  
Hassan Abd Salman Al-Dujaly ◽  
Talat Jassim Aldhlki
Keyword(s):  
Blow Up ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Dirichlet Problems ◽  
Blow Up Rate

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

The Influence of Dirichlet Boundary Conditions on the Dynamics for a Diffusive Predator–Prey System

2019 ◽  
Vol 29 (09) ◽  
pp. 1950113
Author(s):  
Jun Jiang ◽  
Jinfeng Wang ◽  
Yingwei Song
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Linear Instability ◽  
Dirichlet Boundary Conditions ◽  
Homogeneous State ◽  
Predator Prey ◽  
Prey System ◽  
Homogeneous Dirichlet Boundary Conditions

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

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