Non-local Reaction Diffusion Problem Research Articles

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

Nonlinear nonlocal reaction-diffusion problem with local reaction

<p style='text-indent:20px;'>In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.</p>

Evolutionary Branching via Replicator–Mutator Equations

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schr\"odinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur.

Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in R N ( N ≥ 3 )

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem in R N for N≥3 under nonlinear boundary conditions. Utilizing the technique of a differential inequality, lower bounds for the blow-up time are derived when the blow-up does occur under some suitable assumptions.MSC: 35K20, 35K55, 35K65.

Explicit Solutions for Replicator-Mutator Equations: Extinction versus Acceleration

We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.

Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient

Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition are investigated. We derive the conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and obtain an upper bound for . Also we give the condition for global existence of the solution.

Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem under nonlinear boundary conditions. Lower bounds for blow-up time are derived when the blow-up does occur.

On a nonlocal moving frame approximation of traveling waves

The profiles of traveling wave solutions of a 1-d reaction–diffusion parabolic equation are transformed into equilibria of a nonlocal equation, by means of an appropriate nonlocal change of variables. In this new formulation both the profile and the propagation speed of the traveling waves emerge as asymptotic limits of solutions of a nonlocal reaction–diffusion problem when time goes to infinity. In this Note we make these results rigorous analyzing the well-posedness and the stability properties of the corresponding nonlocal Cauchy problem. We also analyze its restriction to a finite interval with consistent boundary conditions. For large enough intervals we show that there is an asymptotically stable equilibrium which approximates the profile of the traveling wave in R . This leads to efficient numerical algorithms for computing the traveling wave profile and speed of propagation.

Lower bounds for the blow-up time in a non-local reaction–diffusion problem

For a non-local reaction–diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived.

Extinction and non-extinction of solutions for a nonlocal reaction-diffusion problem

We investigate extinction properties of solutions for the homogeneous Dirichlet bound- ary value problem of the nonlocal reaction-diffusion equationut−du+ku p = R u q (x,t)dx with p,q ∈ (0,1) and k,d > 0. We show that q = p is the critical extinction exponent. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived.

GLOBAL BLOWUP AND BLOWUP RATE ESTIMATES OF SOLUTIONS FOR A CLASS OF NONLINEAR NON-LOCAL REACTION-DIFFUSION PROBLEMS

In this paper, the global blowup properties of solutions for a class of nonlinear non-local reaction-diffusion problems are investigated by the methods of the prior estimates. Moreover, the blowup rate estimate of the solution is given.

Properties of Positive Solutions for Non-local Reaction-Diffusion Problems

In this paper we investigate the properties of positive solutions for three non-local reaction-diffusion problems. The conditions on the existence and non-existence of global positive solutions are given. Moreover, we prove that the blow-up set is the whole region when the non-linearity occurs in the equation.

Blow-up in a System of Partial Differential Equations with Conserved First Integral. Part II: Problems with Convection

A reaction-diffusion-convection equation with a nonlocal term is studied; the nonlocal operator acts to conserve the spatial integral of the unknown function as time evolves. The equations are parameterised by µ, and for µ = 1 the equation arises as a similarity solution of the Navier-Stokes equations and the nonlocal term plays the role of pressure. For µ = 0, the equation is a nonlocal reaction-diffusion problem. The aim of the paper is to determine for which values of the parameter µ blow-up occurs and to study its form. In particular, interest is focused on the three cases µ 1/2, and µ → 1. It is observed that, for any 0 ≤ µ ≤ 1/2, nonuniform global blow-up occurs; if 1/2 < µ < 1, then the blow-up is global and uniform, while for µ = 1 (the Navier-Stokes equations) there are exact solutions with initial data of arbitrarily large L_∞, L_2, and H^1 norms that decay to zero. Furthermore, one of these exact solutions is proved to be nonlinearly stable in L_2 for arbitrarily large supremum norm. An understanding of this transition from blow-up behaviour to decay behaviour is achieved by a combination of analysis, asymptotics, and numerical techniques.

Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory

In a nonlocal reaction-diffusion model in combustion theory the reaction function involves a physical parameter σ which is a measure of the strength of the reaction mechanism. The purpose of this paper is to show the existence of a critical value σ ∗ such that for σ < σ ∗ a unique global time-dependent solution exists and converges to a steady-state solution as t → ∞, and for σ > σ ∗ the solution blows-up in finite time. A characterization as well as upper and lower bounds of σ ∗ are given.