Bifurcation points for a reaction-diffusion system with two inequalities

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type which is subject to diffusion-driven instability. We show that obstacles (e.g. a unilateral membrane) for both quantities modeled in terms of inequalities introduce a new bifurcation of spatially non-homogeneous steady states in the domain of stability of the trivial solution of the corresponding classical problem without obstacles.     

Upper semicontinuity of the global attractor for the Gierer-Meinhardt model

We are concerned with the following Gierer-Meinhardt model on a bounded domain with smooth boundary ∂Ω which is a biological pattern formation model proposed by A. Gierer and H. Meinhardt

(GM)     

Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

一类带扩散的聚合物模型的定性分析

本文研究一类带扩散的聚合物模型,并在参数γ＞0和γ＜0两种不同情形给出定性分析.主要运用抛物型方程比较原理和相应常微分方程的结论得到反应扩散方程的先验估计和渐近性分析.     

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.     

Feedback Stabilization for a Reaction-Diffusion System with Nonlocal Reaction Term

We consider a two-component reaction-diffusion system with a nonlocal reaction term. A necessary condition and a sufficient condition for the internal stabilizability to zero of one of the two components of the solution while preserving the nonnegativity of both components have been established in [6 S. Ani?a , W.-E. Fitzgibbon , and M. Langlais ( 2009 ). Global existence and internal stabilization for a class of predator-prey systems posed on non coincident spatial domains . Discrete Cont. Dynam. Syst. B 11 : 805 – 822 .[Crossref], [Web of Science ®] , [Google Scholar]]. In case of stabilizability, a feedback stabilizing control of harvesting type has been indicated. The rate of stabilization (for the indicated feedback control) is given by the principal eigenvalue of a certain non-selfadjoint operator. A large principal eigenvalue leads to a fast stabilization. The first main goal of this article is to approximate this principal eigenvalue. This is done in two steps. First, we investigate the large-time behavior of the solution to a logistic population dynamics with migration, and next we derive as a consequence a method to approximate the principal eigenvalue. The other main goal is to derive a conceptual iterative algorithm to improve the position of the support of the control in order to get a faster stabilization. Our results apply to prey-predator systems.     

A Reaction-diffusion System of a Prey with Three Genotypes and a Predator

The global existence and the asymptotic behavior of solutions to a reaction-diffusion system of a prey with three genotypes and a predator are considered. We establish the evolvement of a pure strain. Here an assumption concerning the diffusion is needed.     

Asymptotic behavior of solutions for some chemical reactions models

1.IntroductionInthispapertweareconcernedwiththeasymptoticbehaviorofglobalpositivesolutionstothefollowingsemilinearreaction-dictionsystemac--didn=--any bltv,(1.1)yi--daal=--any bZw,xEfi,t>0(1.2)ac--daal=any--bZw,(1.3)withhomogeneousNeumannboundaryconditionsonofof~~on--an=on,xeoff,t>o(1.4)andinitialvalueconditionsu(x,0)="o(x),v(x,0)=v000,w(x,0)=w000,xEn,(l.5)wherenCRNisaboundeddomainwithsmoothboundaloff,kdenotestheoutwardnormalderiVativeonoff,anddltd2,d3,a,hi,bZarepositiveconstallts.Problem(…     

Time-space white noise eliminates global solutions in reaction-diffusion equations

We prove that perturbing the reaction-diffusion equation u_t=u_xx+(u₊)^p (p>1), with time-space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.