Instability induced by cross-diffusion in reaction–diffusion systems

In this paper the instability of the uniform equilibrium of a general strongly coupled reaction–diffusion is discussed. In unbounded domain and bounded domain the sufficient conditions for the instability are obtained respectively. The conclusion is applied to the ecosystem, it is shown that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion–reaction system.     

Reaction–diffusion in viscoelastic materials

In this paper we study initial boundary value problems that describe reaction–diffusion phenomena in viscoelastic materials. The mathematical model, represented by an integro-differential equation coupled with an ordinary differential equation, is analyzed from theoretical and numerical viewpoints.     

Stationary solutions of non-autonomous Maxwell–Dirac systems

The Maxwell–Dirac system describes the interaction of a particle with its self-generated electromagnetic field. In this paper, we study the existence of least energy stationary solutions for non-autonomous Maxwell–Dirac system with a nonlinear term in (3+1)$(3+1)$-Minkowski space–time via variational methods. This problem is strongly indefinite and presents a lack of compactness. To overcome these difficulties, we will use the linking and concentration compactness arguments.     

Spatial segregation limit of a non-autonomous competition–diffusion system

This paper is concerned with the spatial behavior of the non-autonomous competition–diffusion system arising in population ecology. The limiting profile of the system is given as the competition rate tends to infinity. Our result shows that two competing species spatially segregate as the competition rates become large. Moreover, for the case of the same non-autonomous terms, we obtain the uniform convergence result.     

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.      8. Blow-up rates for semi-linear reaction–diffusion systems      Henghui Zou 《Journal of Differential Equations》2014       9. Stability of mass action reaction–diffusion systems      《Nonlinear Analysis: Theory, Methods & Applications》2004,56(8):1105-1131 Results on stability of two types of chemical reactions, one represented by an acyclic graph and the other as a reversible reaction have been extended to the case of reaction–diffusion systems. Lyapunov functions are used as the major method for showing asymptotic stability of spatially homogeneous equilibria. Some examples are considered for illustration.      10. Dead cores in nonlinear reaction–diffusion systems      《Nonlinear Analysis: Theory, Methods & Applications》1987,11(10):1219-1227       11. A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations      Tongjun Sun  Keying Ma 《Computational Optimization and Applications》2018,71(1):273-306 In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.      12. Critical and bifurcation points of reaction–diffusion systems with conditions given by inclusions      《Nonlinear Analysis: Theory, Methods & Applications》2001,46(1):69-90       13. Note on the stability of reaction–diffusion systems with delays by Lyapunov functional      《Applied Mathematics Letters》2018 This paper aims to investigate the stability of reaction–diffusion equations with delays. We extend a stability theorem on FDEs introduced by Hale to reaction–diffusion equations with time delay.      14. N-species non-autonomous Lotka–Volterra competitive systems with delays and impulsive perturbations      Long Zhang  Zhidong Teng 《Nonlinear Analysis: Real World Applications》2011,12(6):3152-3169       15. 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.      16. Random flights governed by Klein–Gordon-type partial differential equations      Roberto Garra  Enzo Orsingher 《Stochastic Processes and their Applications》2014       17. Slope modulation of waves governed by sine–Gordon equation      Khanh Chau Le  Lu Trong Khiem Nguyen 《Communications in Nonlinear Science & Numerical Simulation》2013,18(7):1563-1567 Using the variational-asymptotic method we develop the theory of slope modulation of wave packet governed by sine–Gordon equation. A class of asymptotic solutions to the equation of slope modulation is found in terms of the density of solitons. The comparison with the exact n-soliton solution of sine–Gordon equation shows quite excellent agreement.      18. Fast–slow diffusion systems with nonlinear boundary conditions      《Nonlinear Analysis: Theory, Methods & Applications》2001,46(6):893-908       19. On ground state solutions for some non-autonomous Schrödinger–Poisson systems      Juntao Sun  Haibo Chen  Juan J. Nieto 《Journal of Differential Equations》2012,252(5):3365-3380       20. Existence of periodic travelling wave solutions of non-autonomous reaction–diffusion equations with lambda–omega type      Shao Yuan Huang  Sui Sun Cheng 《Journal of Mathematical Analysis and Applications》2014 Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ–ω$\lambda \text{–}\omega$ type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ–ω$\lambda \text{–}\omega$ systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.

Stability of mass action reaction–diffusion systems

Results on stability of two types of chemical reactions, one represented by an acyclic graph and the other as a reversible reaction have been extended to the case of reaction–diffusion systems. Lyapunov functions are used as the major method for showing asymptotic stability of spatially homogeneous equilibria. Some examples are considered for illustration.     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

Note on the stability of reaction–diffusion systems with delays by Lyapunov functional

This paper aims to investigate the stability of reaction–diffusion equations with delays. We extend a stability theorem on FDEs introduced by Hale to reaction–diffusion equations with time delay.     

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.     

Slope modulation of waves governed by sine–Gordon equation

Using the variational-asymptotic method we develop the theory of slope modulation of wave packet governed by sine–Gordon equation. A class of asymptotic solutions to the equation of slope modulation is found in terms of the density of solitons. The comparison with the exact n-soliton solution of sine–Gordon equation shows quite excellent agreement.     

Existence of periodic travelling wave solutions of non-autonomous reaction–diffusion equations with lambda–omega type

Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ–ω$\lambda \text{–}\omega$ type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ–ω$\lambda \text{–}\omega$ systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.