Monotonicity and Stability for Some Reaction-Diffusion Systems with Delay and Dirichlet Boundary Conditions

SynopsisWe consider three models of multiple-step combustion processes on bounded spatial domains. Previously, steady-state convergence results have been established for these models with zero Neumann boundary conditions imposed on the temperature as well as the mass fractions. We retain here throughout the same boundary conditions on the mass fractions, but in our first set of results we establish steady-state convergence results with fixed Dirichlet boundary conditions on the temperature. Next, under certain physically reasonable assumptions, we develop, for two of the models, estimates on the decay rates of both mass fractions to zero, while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction. These results hold under either set of boundary conditions, but when the Dirichlet conditions are imposed on the temperature, we are able to obtain estimates on the rate of convergence of the temperature to its (generally nonconstant) steady-state. Finally, we improve the results of a previous paper by adding a temperature convergence result.

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Global-in-time behavior of weak solutions to reaction–diffusion systems with inhomogeneous Dirichlet boundary condition

Nonlinear Analysis ◽

10.1016/j.na.2017.01.013 ◽

2017 ◽

Vol 159 ◽

pp. 393-407 ◽

Cited By ~ 3

Author(s):

Michel Pierre ◽

Takashi Suzuki ◽

Haruki Umakoshi

Keyword(s):

Boundary Condition ◽

Weak Solutions ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Reaction Diffusion ◽

Time Behavior ◽

Reaction Diffusion Systems ◽

Diffusion Systems

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Existence and Global Exponential Stability of Equilibrium Solution to Reaction-Diffusion Recurrent Neural Networks on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2010/624619 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Kaihong Zhao ◽

Yongkun Li

Keyword(s):

Neural Networks ◽

Boundary Conditions ◽

Exponential Stability ◽

Time Scales ◽

Recurrent Neural Networks ◽

Global Exponential Stability ◽

Equilibrium Solution ◽

Dirichlet Boundary ◽

Reaction Diffusion ◽

Dirichlet Boundary Conditions

The existence of equilibrium solutions to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales is proved by the topological degree theory and M-matrix method. Under some sufficient conditions, we obtain the uniqueness and global exponential stability of equilibrium solution to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales by constructing suitable Lyapunov functional and inequality skills. One example is given to illustrate the effectiveness of our results.

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Reaction-advection-diffusion competition models under lethal boundary conditions

Discrete and Continuous Dynamical Systems - Series B ◽

10.3934/dcdsb.2021250 ◽

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>

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