Scalar Reaction-diffusion Equations Research Articles

On the structure of the infinitesimal generators of scalar one-dimensional semigroups with discrete Lyapunov functionals

AbstractDynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $$C^0[0,1]$$ C 0 [ 0 , 1 ] or on $$C^1[0,1]$$ C 1 [ 0 , 1 ] which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $$C^1[0,1]$$ C 1 [ 0 , 1 ] , the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $$C^0[0,1]$$ C 0 [ 0 , 1 ] , the corresponding equations are generalizations of monotone feedback delay differential equations.

Continuity of the Unbounded Attractors for a Fractional Perturbation of a Scalar Reaction-Diffusion Equation

Continuity of the Unbounded Attractors for a Fractional Perturbation of a Scalar Reaction-Diffusion Equation

The role of convection in the existence of wavefronts for biased movements

We investigate a model, inspired by Johnston et al. (2017) to describe the movement of a biological population which consists of isolated and grouped organisms. We introduce biases in the movements and then obtain a scalar reaction–diffusion equation that includes a convective term as a consequence of the biases. We focus on the case the diffusivity makes the parabolic equation of forward–backward–forward type and the reaction term models a strong Allee effect, with the Allee parameter lying between the two internal zeros of the diffusion. In such a case, the unbiased equation (i.e., without convection) possesses no smooth traveling‐wave solutions; on the contrary, in the presence of convection, we show that traveling‐wave solutions do exist for some significant choices of the parameters. We also study the sign of their speeds, which provides information on the long term behavior of the population, namely, its survival or extinction.

Non-wandering points for autonomous/periodic parabolic equations on the circle

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle S 1 , u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , where f is independent of t or T -periodic in t . Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some ω -limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if f ( t , u , − u x ) = f ( t , u , u x ) , then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.

Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations II: HHO-Inspired Methods

In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree \(k\geqslant 1\). In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for \(k\geqslant 0\) by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

An Exact Solution of the Reaction-Diffusion Equation for the Speed of the Interface Propagation in Superconductors

The speed of interface propagation in superconductors for the scalar reaction-diffusion equation ut  =   ∇2u+ F(u)   is studied in detail. Here the non linear reaction term F (u) is the time-dependent Ginzburg-Landau or TDGL equation F(u)=u-u3 which describes the dynamics of the order-disorder transition. In contrast to what has been done in previous work [1] here an improved exact solution has derived by using TDGL equation to determine the speed of the front propagation. The analytical treatment of this study has been found in good agreement with the numerical simulation of V. Mendez et al. [2] and Di Bartolo and Dorsey [3].
 The Chittagong Univ. J. Sci. 40(1) : 85-95, 2019

A minimizing Movement approach to a class of scalar reaction–diffusion equations

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form [see formula in PDF] with parameters Λ, Σ > 0 and no-flux boundary condition [see formula in PDF] which is built on their gradient-flow-like structure in the space [see formula in PDF] of finite nonnegative Radon measures on [see formula in PDF], endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on [see formula in PDF] and [see formula in PDF], the Minimizing Movement scheme [see formula in PDF] for [see formula in PDF] yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.

Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization.

Many monostable reaction-diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction-diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction-diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction-diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

Competition in Periodic Media: III—Existence and Stability of Segregated Periodic Coexistence States

In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We are mainly concerned with the case of periodic coefficients and periodic solutions. We show that, for sufficiently large periods, these models have stationary, non-constant, fully non-trivial and stable solutions. We compare our results with already known results about the existence and non-existence of such solutions. Finally, we provide ecological interpretations for these results in terms of resistance against an invasion.

Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

In this paper, we consider the scalar reaction–diffusion equations $$\partial _t u= \Delta u+f(x,u, \nabla u)$$ on a bounded domain $$\Omega \subset \mathbb {R}^d$$ of class $$\mathcal {C}^{2,\gamma }$$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka–Smale property, the generic hyperbolicity of periodic orbits remaining unproved.

Nonstandard existence proofs for reaction diffusion equations

We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness theorems, such as Ascoli's theorem and the Banach--Alaoglu theorem on weak$^*$-compactness of the unit ball in the dual of a Banach space.

A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 2: The 2-D case

This is the second part to our companion paper [18]. Herein, we generalize to two space dimensions the C-method developed in [20,18] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities. For gas dynamics, the C-method couples the Euler equations to scalar reaction-diffusion equations, which we call C-equations, whose solutions serve as space-time smooth artificial viscosity indicators for shocks and contacts.We develop a high-order numerical algorithm for gas dynamics in 2-D which can accurately simulate the Rayleigh-Taylor (RT) instability with Kelvin-Helmholtz (KH) roll-up of the contact discontinuity, as well as shock collision and bounce-back. Solutions to our C-equations not only indicate the location of the shocks and contacts, but also track the geometry of the evolving fronts. This allows us to implement both directionally isotropic and anisotropic artificial viscosity schemes, the latter adding diffusion only in directions tangential to the evolving front. We additionally produce a novel shock collision indicator function, which naturally activates during shock collision, and then smoothly deactivates. Moreover, we implement a high-frequency 2-D wavelet-based noise detector together with an efficient and localized noise removal algorithm.To test the methodology, we use a highly simplified WENO-based discretization scheme. We provide numerical results for some classical 2-D test problems, including the RT problem, the Noh problem, a circular explosion problem from the Liska & Wendroff [13] review paper, the Sedov blast wave problem, the double Mach 10 reflection test, and a shock-wall collision problem. In particular, we show that our artificial viscosity method can eliminate the wall-heating phenomenon for the Noh problem, and thereby produce an accurate, non-oscillatory solution, even though our simplified WENO-type scheme fails to run for this problem.

A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: The 1-D case

In this first part of two papers, we extend the C-method developed in [40] for adding localized, space-time smooth artificial viscosity to nonlinear systems of conservation laws that propagate shock waves, rarefaction waves, and contact discontinuities in one space dimension. For gas dynamics, the C-method couples the Euler equations to a scalar reaction-diffusion equation, whose solution C serves as a space-time smooth artificial viscosity indicator.The purpose of this paper is the development of a high-order numerical algorithm for shock-wall collision and bounce-back. Specifically, we generalize the original C-method by adding a new collision indicator, which naturally activates during shock-wall collision. Additionally, we implement a new high-frequency wavelet-based noise detector together with an efficient and localized noise removal algorithm. To test the methodology, we use a highly simplified WENO-based discretization scheme. We show that our scheme improves the order of accuracy of our WENO algorithm, handles extremely strong discontinuities (ranging up to nine orders of magnitude), allows for shock collision and bounce back, and removes high frequency noise. The causes of the well-known “wall heating” phenomenon are discussed, and we demonstrate that this particular pathology can be effectively treated in the framework of the C-method. This method is generalized to two space dimensions in the second part of this work [41].

Entire solutions originating from monotone fronts to the Allen–Cahn equation

In this paper, we study entire solutions of the Allen–Cahn equation in one-dimensional Euclidean space. This equation is a scalar reaction–diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions originating from three and four fronts is shown by constructing some suitable pairs of super–sub-solutions. Moreover, we show that there are no entire solutions originating from more than four fronts.

Instability of pulses in gradient reaction-diffusion systems: a symplectic approach.

In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have non-zero Maslov index, and hence be unstable.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the elliptic PDE $\begin{cases} \partial_t^2 u - c \partial_t u + \Delta u + f(x,u) = 0, \qquad & t \in \mathbf{R},\; x \in \Omega, \\ B(u) = 0, & t \in \mathbf{R},\; x \in \partial\Omega, \end{cases}$ where $c \neq 0$ is the wave speed, $\Omega \subset \mathbf{R}^d$ is a bounded domain, $\Delta$ is the Laplacian on $\Omega$, and $B$ denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. Restrictions on the nonlinearity $f$ are kept to a minimum; for instance, any nonlinearity exhibiting polynomial growth in $u$ can be considered. In particular, the set of bounded solutions of the traveling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity $f$, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.

Polarization of concave domains by traveling wave pinning.

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

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.

Hopf bifurcation in a reaction–diffusion equation with distributed delay and Dirichlet boundary condition

The stability and Hopf bifurcation of the positive steady state to a general scalar reaction–diffusion equation with distributed delay and Dirichlet boundary condition are investigated in this paper. The time delay follows a Gamma distribution function. Through analyzing the corresponding eigenvalue problems, we rigorously show that Hopf bifurcations will occur when the shape parameter n≥1, and the steady state is always stable when n=0. By computing normal form on the center manifold, the direction of Hopf bifurcation and the stability of the periodic orbits can also be determined under a general setting. Our results show that the number of critical values of delay for Hopf bifurcation is finite and increasing in n, which is significantly different from the discrete delay case, and the first Hopf bifurcation value is decreasing in n. Examples from population biology and numerical simulations are used to illustrate the theoretical results.

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut=uxx+α[βH(u−θ)−u]−w, wt=ε(u−γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u], under different conditions on the model constants.To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0<2(1+αγ)θ<αβγ; the existence and stability of two standing wave fronts if 2(1+αγ)θ=αβγ and γ2ε>1; the existence and instability of two standing wave fronts if 2(1+αγ)θ=αβγ and 0<γ2ε<1; the existence and instability of an upside down standing pulse solution if 0<(1+αγ)θ<αβγ<2(1+αγ)θ. To establish the bifurcation for the scalar equation, we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0<2θ<β; the existence and stability of two standing wave fronts if 2θ=β; the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0<θ<β<2θ. By the way, we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation ut=uxx+α[βH(u−θ)−u]−w0, where w0=α(β−2θ)>0 is a positive constant, if 0<2θ<β.To achieve the main goals, we will make complete use of the special structures of the model equations and we will construct Evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators. It turns out that a complex number λ0 is an eigenvalue of the linear differential operator, if and only if λ0 is a zero of the Evans function. The stability, instability and bifurcations of the nonlinear waves follow from the zeros of the Evans functions.A very important motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.