Long-term coexistence for a competitive system of spatially varying gradient reaction–diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction–diffusion equations. In this work we consider a system of two reaction–diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations.     

Finite difference reaction–diffusion equations with nonlinear diffusion coefficients

Summary. A monotone iterative method for numerical solutions of a class of finite difference reaction-diffusion equations with nonlinear diffusion coefficient is presented. It is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system. It is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero. Received February 18, 1998 / Revised version received April 21, 1999 / Published online February 17, 2000     

Global stability analysis for stochastic coupled reaction–diffusion systems on networks

Coupled systems on networks (CSNs) can be used to model many real systems, such as food webs, ecosystems, metabolic pathways, the Internet, World Wide Web, social networks, and global economic markets. This paper is devoted to investigation of the stability problem for some stochastic coupled reaction–diffusion systems on networks (SCRDSNs). A systematic method for constructing global Lyapunov function for these SCRDSNs is provided by using graph theory. The stochastic stability, asymptotically stochastic stability and globally asymptotically stochastic stability of the systems are investigated. The derived results are less conservative than the results recently presented in Luo and Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493]. In fact, the system discussed in Q. Luo and Y. Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493] is a special case of ours. Moreover, our novel stability principles have a close relation to the topological property of the networks. Our new method which constructs a relation between the stability criteria of a CSN and some topology property of the network, can help analyzing the stability of the complex networks by using the Lyapunov functional method.     

Uniqueness of transverse solutions for reaction–diffusion equations with spatially distributed hysteresis

The paper deals with reaction–diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. Such problems describe biological processes and chemical reactions in which diffusive and nondiffusive substances interact according to hysteresis law. Under the assumption that the initial data are spatially transverse, we prove a theorem on the uniqueness of solutions. The theorem covers the case of non-Lipschitz hysteresis branches arising in the theory of slow–fast systems.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. Blow-up rates for semi-linear reaction–diffusion systems      Henghui Zou 《Journal of Differential Equations》2014       8. Traveling waves for a boundary reaction–diffusion equation      L. Caffarelli  A. Mellet  Y. Sire 《Advances in Mathematics》2012,230(2):433-457 We prove the existence of a traveling wave solution for a boundary reaction–diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction–diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).      9. Higher-order integrability for a semilinear reaction–diffusion equation with distribution derivatives in RN      Chunyou Sun  Lili Yuan  Jiancheng Shi 《Applied Mathematics Letters》2013,26(9):949-956       10. Fast–slow diffusion systems with nonlinear boundary conditions      《Nonlinear Analysis: Theory, Methods & Applications》2001,46(6):893-908       11. Invariant sets and the blow up threshold for coupled systems of reaction–diffusion      Baiyu Liu  Xinhui Si 《Applicable analysis》2013,92(4):637-652 In this paper, we study a class of semilinear systems of reaction–diffusion on a bounded smooth domain with Dirichlet boundary condition. Applying the potential well method, we find invariant sets for the initial-boundary value problem and derive a threshold of blow up and global existence for its solution.      12. First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations      《Applied Mathematics Letters》2019 We suggest in this letter a new Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion–reaction equations where the random walk is living both on the randomly walking spheres and inside the relevant balls. The method is mesh free both in space and time, and is well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Some applications to exciton flux calculations in the diffusion imaging method in semiconductors are discussed.      13. Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions      Milan Kučera  Martin Väth 《Journal of Differential Equations》2012,252(4):2951-2982 We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.      14. Global existence for degenerate quadratic reaction–diffusion systems      M. Pierre  R. Texier-Picard 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009 We consider a class of degenerate reaction–diffusion systems with quadratic nonlinearity and diffusion only in the vertical direction. Such systems can appear in the modeling of photochemical generation and atmospheric dispersion of pollutants. The diffusion coefficients are different for all equations. We study global existence of solutions.      15. Traveling waves for a reaction–diffusion–advection system with interior or boundary losses      Thomas Giletti 《Comptes Rendus Mathematique》2011,349(9-10):535-539 This Note deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction–diffusion system with losses inside the domain. In particular, we show the existence of a continuum of admissible speeds of traveling waves. Lastly, by considering losses concentrated near the boundary of the domain, these results are compared with those already known in the case of losses on the boundary.      16. Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth      Belgacem Rebiai  Saïd Benachour 《Journal of Evolution Equations》2010,10(3):511-527 The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in \mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +￥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.      17. On global initial boundary-value problems for reaction–diffusion systems with balance conditions      《Nonlinear Analysis: Theory, Methods & Applications》1999,37(8):971-995       18. Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions      Juntang Ding  Shengjia Li 《Nonlinear Analysis: Theory, Methods & Applications》2008       19. Finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay via periodically intermittent control      Yao Xu  Yongbao Wu 《Applicable analysis》2013,92(9):1660-1676 In this paper, the issue of finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay (CRDSTD) is considered. A periodically intermittent controller is designed such that drive system and corresponding response system can achieve finite-time lag synchronization. By using graph theory and Lyapunov method, two sufficient criteria are presented to guarantee the finite-time lag synchronization of CRDSTD. Moreover, the time of achieving lag synchronization of CRDSTD is estimated. Finally, a numerical example is given to show the effectiveness of the proposed results.      20. Traveling waves of delayed reaction–diffusion systems with applications      Zhi-Xian Yu  Rong Yuan 《Nonlinear Analysis: Real World Applications》2011,12(5):2475-2488

Traveling waves for a boundary reaction–diffusion equation

We prove the existence of a traveling wave solution for a boundary reaction–diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction–diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).     

Invariant sets and the blow up threshold for coupled systems of reaction–diffusion

In this paper, we study a class of semilinear systems of reaction–diffusion on a bounded smooth domain with Dirichlet boundary condition. Applying the potential well method, we find invariant sets for the initial-boundary value problem and derive a threshold of blow up and global existence for its solution.     

First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations

We suggest in this letter a new Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion–reaction equations where the random walk is living both on the randomly walking spheres and inside the relevant balls. The method is mesh free both in space and time, and is well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Some applications to exciton flux calculations in the diffusion imaging method in semiconductors are discussed.     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

Global existence for degenerate quadratic reaction–diffusion systems

We consider a class of degenerate reaction–diffusion systems with quadratic nonlinearity and diffusion only in the vertical direction. Such systems can appear in the modeling of photochemical generation and atmospheric dispersion of pollutants. The diffusion coefficients are different for all equations. We study global existence of solutions.     

Traveling waves for a reaction–diffusion–advection system with interior or boundary losses

This Note deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction–diffusion system with losses inside the domain. In particular, we show the existence of a continuum of admissible speeds of traveling waves. Lastly, by considering losses concentrated near the boundary of the domain, these results are compared with those already known in the case of losses on the boundary.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay via periodically intermittent control

In this paper, the issue of finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay (CRDSTD) is considered. A periodically intermittent controller is designed such that drive system and corresponding response system can achieve finite-time lag synchronization. By using graph theory and Lyapunov method, two sufficient criteria are presented to guarantee the finite-time lag synchronization of CRDSTD. Moreover, the time of achieving lag synchronization of CRDSTD is estimated. Finally, a numerical example is given to show the effectiveness of the proposed results.