Asymptotic behaviour of some reaction-diffusion systems modelling complex combustion on bounded domains

1993 ◽  
Vol 123 (6) ◽  
pp. 1151-1163
Author(s):  
Joel D. Avrin
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Convergence Result ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Systems ◽  
Convergence Results ◽  
Mass Fractions

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.

scholarly journals Synchronization for a class of generalized neural networks with interval time-varying delays and reaction-diffusion terms

2016 ◽  
Vol 21 (3) ◽  
pp. 379-399 ◽  
Author(s):  
Qintao Gan ◽  
Tielin Liu ◽  
Chang Liu ◽  
Tianshi Lv
Keyword(s):  
Neural Networks ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Dirichlet Boundary Conditions ◽  
Linear Matrix ◽  
Time Varying ◽  
Interval Time ◽  
Generalized Neural Networks ◽  
Special Cases

In this paper, the synchronization problem for a class of generalized neural networks with interval time-varying delays and reaction-diffusion terms is investigated under Dirichlet boundary conditions and Neumann boundary conditions, respectively. Based on Lyapunov stability theory, both delay-derivative-dependent and delay-range-dependent conditions are derived in terms of linear matrix inequalities (LMIs), whose solvability heavily depends on the information of reaction-diffusion terms. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. The obtained synchronization results are easy to check and improve upon the existing ones. In our results, the assumptions for the differentiability and monotonicity on the activation functions are removed. It is assumed that the state delay belongs to a given interval, which means that the lower bound of delay is not restricted to be zero. Finally, the feasibility and effectiveness of the proposed methods is shown by simulation examples.

scholarly journals Effective rough boundary parametrization for reaction-diffusion systems

2014 ◽  
Vol 8 (1) ◽  
pp. 33-59 ◽  
Author(s):  
C. Mocenni ◽  
E. Sparacino ◽  
J.P. Zubelli
Keyword(s):  
Boundary Conditions ◽  
Periodic Structures ◽  
Reaction Diffusion ◽  
Multiple Scale ◽  
Dirichlet Boundary Conditions ◽  
Homogenization Theory ◽  
Rough Boundary ◽  
Dynamical Regime ◽  
Effective Equation ◽  
Reaction Diffusion Systems

We address the problem of parametrizing the boundary data for reaction- diffusion partial differential equations associated to distributed systems that possess rough boundaries. The boundaries are modeled as fast oscillating periodic structures and are endowed with Neumann or Dirichlet boundary conditions. Using techniques from homogenization theory and multiple-scale analysis we derive the effective equation and boundary conditions that are satisfied by the homogenized solution. We present numerical simulations that validate our theoretical results and compare it with the alternative approach based on solving the same equation with a smoothed version of the boundary. The numerical tests show the accuracy of the homogenized solution to the effective system vis a vis the numerical solution of the original differential equation. The homogenized solution is shown undergoing dynamical regime shifts, such as anticipation of pattern formation, obtained by varying the diffusion coefficient.

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.

Spike-Layer Simulation for Steady-State Coupled Schrödinger Equations

2017 ◽  
Vol 7 (3) ◽  
pp. 566-582 ◽  
Author(s):  
Liming Liao ◽  
Guanghua Ji ◽  
Zhongwei Tang ◽  
Hui Zhang
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Circular Ring ◽  
Dirichlet Boundary Conditions ◽  
Adaptive Finite Element Method ◽  
Schrodinger Equations ◽  
Adaptive Finite Element ◽  
Schrödinger Equations ◽  
Nonlinear Algebraic System

AbstractAn adaptive finite element method is adopted to simulate the steady state coupled Schrödinger equations with a small parameter. We use damped Newton iteration to solve the nonlinear algebraic system. When the solution domain is elliptic, our numerical results with Dirichlet or Neumann boundary conditions are consistent with previous theoretical results. For the dumbbell and circular ring domains with Dirichlet boundary conditions, we obtain some new results that may be compared with future theoretical analysis.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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