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.     

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.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.     

An approximate solution of nonlinear fractional reaction–diffusion equation

The article presents a mathematical model of nonlinear reaction diffusion equation with fractional time derivative α (0 < α ? 1) in the form of a rapidly convergent series with easily computable components. Fractional reaction diffusion equation is used for modeling of merging travel solutions in nonlinear system for popular dynamics. The fractional derivatives are described in the Caputo sense. The anomalous behaviors of the nonlinear problems in the form of sub- and super-diffusion due to the presence of reaction term are shown graphically for different particular cases.     

The comparison between the DRBEM and DQM solution of nonlinear reaction–diffusion equation

In this study, both the dual reciprocity boundary element method and the differential quadrature method are used to discretize spatially, initial and boundary value problems defined by single and system of nonlinear reaction–diffusion equations. The aim is to compare boundary only and a domain discretization method in terms of accuracy of solutions and computational cost. As the time integration scheme, the finite element method is used achieving solution in terms of time block with considerably large time steps. The comparison between the dual reciprocity boundary element method and the differential quadrature method solutions are made on some test problems. The results show that both methods achieve almost the same accuracy when they are combined with finite element method time discretization. However, as a method providing very good accuracy with considerably small number of grid points differential quadrature method is preferrable.     

Conservation laws for a variable coefficient nonlinear diffusion–convection–reaction equation

We find the Lie point symmetries of a class of second-order nonlinear diffusion–convection–reaction equations containing an unspecified coefficient function of the independent variable t and determine the subclasses of these equations which are nonlinearly self-adjoint. By using a general theorem on conservation laws proved recently by N.H. Ibragimov we establish conservation laws corresponding to the aforementioned Lie point symmetries, one by one, for the simultaneous system of the original equation together with its adjoint equation through a formal Lagrangian. Particularly, for the nonlinearly self-adjoint subclasses, we construct conservation laws for the corresponding equations themselves.     

Blow-up for a reaction–diffusion equation with variable coefficient

We study the blow-up behavior for positive solutions of a reaction–diffusion equation with nonnegative variable coefficient. When there is no stationary solution, we show that the solution blows up in finite time. Under certain conditions, we then show that any point with zero source cannot be a blow-up point.     

Singularity analysis of a reaction–diffusion equation with a solution-dependent Dirac delta source

We analyze the existence and singularity of a solution to a reaction–diffusion equation, whose reaction term is represented by a Dirac delta function which depends on the solution itself. We prove that there exists a unique analytic solution with a logarithmic singularity at the origin.     

Stability of a nonlocal delayed reaction–diffusion equation with a non-monotone bistable nonlinearity

The present work is devoted to the stability and attractivity analysis of a nonlocal delayed reaction–diffusion equation (DRDE) with a non-monotone bistable nonlinearity that describes the population dynamics for a two-stage species with Allee effect. By the idea of relating the dynamics of the nonlinear term to the DRDE and some stability results for the monostable case, we describe some basin of attractions for the DRDE. Additionally, existence of heteroclinic orbits and periodic oscillations are also obtained. Numerical simulations are also given at last to verify our theoretical results.     

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     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.     

Averaging principle for a class of stochastic reaction–diffusion equations

We consider the averaging principle for stochastic reaction–diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE’s.     

Blow-up and global solutions for a class of time fractional nonlinear reaction–diffusion equation with weakly spatial source

This paper investigates the blow-up of solutions for a time fractional nonlinear reaction–diffusion equation with weakly spatial source. We first derive two sufficient conditions under which the solutions may blow up in finite time. Then, we prove the existence of global solution when the initial data are small enough. Moreover, the long time behavior of bounded solutions will be analyzed.     

Blow-up analysis for a reaction–diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux

A blow-up analysis for a nonlinear divergence form reaction–diffusion equation with weighted nonlocal inner absorption terms is considered under nonlinear boundary flux in a bounded star-shaped region. Based on the auxiliary function method and the modified differential inequality technique, we establish conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, upper and lower bounds for the blow-up time are derived under appropriate measure in higher dimensional spaces. Three examples are presented to illustrate applications of our main results.     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.