On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.     

Existence and uniqueness of solutions of reaction–convection equations with non-Lipschitz nonlinearity

In this work we study single balance law _ut+∇⋅Φ(u)=f(u)$u_t+\mathrm{\nabla }\cdot \Phi (u)=f(u)$ with bounded initial value, and find that there may exist maximal and minimal solutions, if f(u)$f(u)$ is not Lipschitz continuous at u=0$u=0$. We also show that comparison principle is valid for such solutions, and the solutions may blow up or not under certain conditions. It is determined by the strength of source supply, as well as the competition between the source and flux.     

Nontrivial large-time behaviour in bistable reaction–diffusion equations

Bistable reaction–diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations—Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations—Xin [30], Levermore-Xin [19], Kapitula [16]—provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection–diffusion equations.      

Systems of reaction–convection–diffusion equations invariant under Galilean algebras

Systems, which are invariant under Galilean algebras (with and without mass operator), and their main extensions by operators of scale and projective transformations are selected from a class of systems of reaction–convection–diffusion equations for two-dimensional vector field with one space variable.     

Oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability

This paper is concerned with the existence of oscillatory waves in reaction–diffusion equations with nonlocal delay and crossing-monostability, which include many population models, and two main results are presented. In the first one, we establish the existence of non-monotone traveling waves from the trivial solution to the positive equilibrium. The approach is based on the construction of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using traveling fronts of the auxiliary equations. In the second one, we obtain the existence of periodic waves around the positive equilibrium by using Hopf bifurcation theorem.     

Travelling wavefronts of scalar reaction–diffusion equations with and without delays

This paper deals with the existence of travelling wavefronts for scalar nonlinear reaction–diffusion equations with and without delays in one-dimensional space. New iterative techniques for a class of integral operators of Hammerstein type are established and applied to tackle the existence of travelling wavefronts in a unified way. Our results without delays only require the functions involved to be continuous and satisfy a suitable monotonicity condition. Our results with multiple delays employ the usual C¹-assumption but generalize the well-known results.     

Spatially discrete reaction–diffusion equations with discontinuous hysteresis

We address the question: Why may reaction–diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order $a{t}^{?1/2}$ as $t\to \infty$ and explicitly find the rate a.     

Time-independent reaction–diffusion equation with a discontinuous reactive term

A two-dimensional singularly perturbed elliptic equation referred to in applications as the reaction–diffusion equation is considered. The nonlinearity describing the reaction is assumed to be discontinuous on a certain closed curve. On the basis of the generalized asymptotic comparison principle, the existence of smooth solution is proven and the accuracy of the asymptotic approximation is estimated.     

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     

Periodic homogenization of strongly nonlinear reaction–diffusion equations with large reaction terms

We study in this article the periodic homogenization problem related to a strongly nonlinear reaction–diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that the homogenized equation is exactly of a convection-diffusion type. This study relies on a suitable version of the well-known two-scale convergence method.     

Periodic solutions in reaction–diffusion equations with time delay

Spatial diffusion and time delay are two main factors in biological and chemical systems. However, the combined effects of them on diffusion systems are not well studied. As a result, we investigate a nonlinear diffusion system with delay and obtain the existence of the periodic solutions using coincidence degree theory. Moreover, two numerical examples confirm our theoretical results. The obtained results can also be applied in other related fields.