Transition fronts in inhomogeneous Fisher–KPP reaction–diffusion equations

We use a new method in the study of Fisher–KPP reaction–diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction–diffusion equations in several spatial dimensions. Our method is based on the construction of sub- and super-solutions to the non-linear PDE from solutions of its linearization at zero.     

Multiplicity of supercritical fronts for reaction–diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.     

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. Fractional amplitude and phase dynamics in super-diffusive reaction–diffusion systems      Y. Nec  A.A. Nepomnyashchy  A.A. Golovin 《PAMM》2007,7(1):2040025-2040026 Study of weakly non-linear dynamics of a reaction–super-diffusion system near a Hopf bifurcation by means of fractional analogues of complex Ginzburg-Landau and Kuramoto-Sivashinsky equations is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)      8. 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.      9. Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition      Ping Liu  Junping Shi 《Journal of Differential Equations》2018,264(1):425-454 The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.      10. Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media      《Nonlinear Analysis: Real World Applications》2022 In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.      11. Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry      Fiammetta Conforto  Laurent Desvillettes  Roberto Monaco 《Ricerche di matematica》2017,66(1):99-111 This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.      12. 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.      13. Bifurcation and stability of a two-species reaction–diffusion–advection competition model      《Nonlinear Analysis: Real World Applications》2021 This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.      14. Asymptotic behavior of pulsating fronts and entire solutions of reaction–advection–diffusion equations in periodic media      《Nonlinear Analysis: Real World Applications》2016       15. An inverse problem involving two coefficients in a nonlinear reaction–diffusion equation      Michel Cristofol  Lionel Roques 《Comptes Rendus Mathematique》2012,350(9-10):469-473       16. Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation      《Applied Mathematical Modelling》2014,38(5-6):1612-1621 Nonlinear reaction–diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction–diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction–diffusion systems, the well-studied Schnakenberg model and chloride–iodide–malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results.      17. A waiting time phenomenon for modulations of pattern in reaction–diffusion systems      Wolf-Patrick?DüllEmail author 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,8(1):1-23 In order to describe slow modulations in time and space of stable or slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, so-called phase diffusion equations and Cahn–Hilliard equations can be derived via multiple scaling analysis as formal approximation equations. In the case that these equations degenerate, waiting time phenomena are well known to occur. In this paper, we prove that such waiting time phenomena can also occur approximately in the original reaction–diffusion systems by proving estimates between the formal approximations and the exact solutions of the original systems.      18. Periodic pyramidal traveling fronts of bistable reaction–diffusion equations with time-periodic nonlinearity      Wei-Jie Sheng  Wan-Tong Li  Zhi-Cheng Wang 《Journal of Differential Equations》2012,252(3):2388-2424       19. Nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equations      Guangying Lv  Mingxin Wang 《Journal of Mathematical Analysis and Applications》2012,385(2):1094-1106       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

Fractional amplitude and phase dynamics in super-diffusive reaction–diffusion systems

Study of weakly non-linear dynamics of a reaction–super-diffusion system near a Hopf bifurcation by means of fractional analogues of complex Ginzburg-Landau and Kuramoto-Sivashinsky equations is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Bifurcation of positive solutions to scalar reaction–diffusion equations with nonlinear boundary condition

The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.     

Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.     

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.     

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.     

Bifurcation and stability of a two-species reaction–diffusion–advection competition model

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.     

Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation

Nonlinear reaction–diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction–diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction–diffusion systems, the well-studied Schnakenberg model and chloride–iodide–malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results.     

A waiting time phenomenon for modulations of pattern in reaction–diffusion systems

In order to describe slow modulations in time and space of stable or slightly unstable spatially periodic stationary solutions of pattern forming reaction–diffusion systems, so-called phase diffusion equations and Cahn–Hilliard equations can be derived via multiple scaling analysis as formal approximation equations. In the case that these equations degenerate, waiting time phenomena are well known to occur. In this paper, we prove that such waiting time phenomena can also occur approximately in the original reaction–diffusion systems by proving estimates between the formal approximations and the exact solutions of the original systems.