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 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.     

Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c∈]_c?,+∞[$c\in ]c_{?},+\infty [$, where _c?>0$c_{?}>0$ is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where u=0$u=0$ and a region where u>0$u>0$, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u>0$u>0$, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.     

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.     

Sixth-order Cahn–Hilliard equations with singular nonlinear terms

Our aim in this paper is to study the well-posedness for a class of sixth-order Cahn–Hilliard equations with singular nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.     

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of $\Sigma$-convergence for stochastic processes.     

Boundary null-controllability of linear diffusion–reaction equations

This Note deals with the boundary null-controllability of linear diffusion–reaction equations in a 2D bounded domain. We transform the determination of the sought HUM boundary control into the minimization of a continuous and strictly convex functional. In the case of a rectangular domain where the diffusion tensor is represented by a diagonal matrix, we establish a procedure based on the inner product method that uses a complete orthonormal family of Sturm–Liouville's eigenfunctions to express explicitly the sought control.     

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.     

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.     

Special features of strongly coupled systems of convection–diffusion equations with two small parameters

Strong coupling of convection–diffusion equations with two small parameters generates a solution decomposition which differs significantly from that for the one-parameter case. We explain the basic features and prove pointwise estimates for the first-order derivatives which allow us to analyze the upwind finite difference scheme on layer-adapted meshes.     

Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations

In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.     

On the solutions of time-fractional reaction–diffusion equations

In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.     

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.     

Qualitative behavior of numerical traveling solutions for reaction–diffusion equations with memory

In this article the qualitative properties of numerical traveling wave solutions for integro- differential equations, which generalize the well known Fisher equation are studied. The integro-differential equation is replaced by an equivalent hyperbolic equation which allows us to characterize the numerical velocity of traveling wave solutions. Numerical results are presented.