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.     

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.     

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.     

Analytical solutions for a time-fractional axisymmetric diffusion–wave equation with a source term

In this paper, a time-fractional axisymmetric diffusion–wave equation with a source term is considered in cylindrical coordinates. The analytical solution is obtained with the help of an integral transform method and some properties of special functions. In addition, we discuss two kinds of different boundary conditions and different forms of the source term. Finally, we analyze the effects of the fractional derivative on the solutions by using numerical results and find that sub-diffusion phenomena and oscillations exist.     

Traveling waves for a boundary reaction–diffusion equation

We prove the existence of a traveling wave solution for a boundary reaction–diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction–diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).     

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.     

On a reaction–advection–diffusion equation with Robin and free boundary conditions

ABSTRACT

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy for the asymptotic behavior of the solutions of the equation. Then, we obtain criteria for spreading and vanishing, and get an estimate for the asymptotic spreading speed of the spreading front. Moreover, numerical simulation is also given to illustrate the impact of the expansion capacity on the free boundary.     

Convergence to a terrace solution in multistable reaction–diffusion equation with discontinuities

In this paper we address the large-time behavior of solutions of bistable and multistable reaction–diffusion equation with discontinuities around the stable steady states. We show that the solution always converges to a special solution, which may either be a traveling wave in the bistable case, or more generally a terrace (i.e. a collection of stacked traveling waves with ordered speeds) in the multistable case.     

Sliding mode control for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation

We consider the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation perturbed by a maximal monotone nonlinearity. We prove existence and regularity of strong solutions and, under further assumptions, a uniqueness result. Then, we show that the chosen SMC law forces the system to reach within finite time a sliding manifold that we chose in order that the tumor phase remains constant in time.     

Solution with an inner transition layer of a two-dimensional boundary value reaction–diffusion–advection problem with discontinuous reaction and advection terms

Theoretical and Mathematical Physics - We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the...     

A skew-symmetric iterative method for solving a steady convection–diffusion–reaction equation with an indefinite reaction coefficient

An iterative product-type triangular skew-symmetric method (PTSM) is used to solve systems of linear algebraic equations (SLAEs) obtained by approximation with a central-difference scheme of a first-type boundary value problem for convection–diffusion–reaction and standard grid ordering. Sufficient conditions for non-negative definiteness of the SLAE matrix resulting from this approximation are obtained for the indefinite reaction coefficient. This property provides convergence of a wide class of iterative methods (in particular, the PTSM). In test problems, agreement of the theory with computational experiments is shown, and a comparison of the PTSM and SSOR is done.     

Speed of wave propagation for a nonlocal reaction–diffusion equation

ABSTRACT

A nonlocal reaction–diffusion equation arising in various applications is studied. The speed of traveling waves is determined by means of a minimax representation. It is used to obtain the wave speed estimates and asymptotic values.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

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.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.