Global existence for coupled reaction-diffusion equations with a balance law and nonlinearities with non-constant sign

This paper aims to prove the global existence of solutions for coupled reaction diffusion equations with a balance Law and nonlinearities with a non constant sign. The case when one (or both) of the components of the solution is not a priori bounded is treated. Proofs are based on developed Lyapunov techniques

A remark on "Study of a Leslie-Gower-type tritrophic population model" [Chaos, Solitons and Fractals 14 (2002) 1275-1293]

In [Aziz-Alaoui, 2002] a three species ODE model, based on a modified Leslie-Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in [Aziz-Alaoui, 2002], if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations.Comment: 10 pages, 4 figure

Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditions via a Lyapunov functional

Most publications on reaction-diffusion systems of $m$ components ($mgeq 2$) impose $m$ inequalities to the reaction terms, to prove existence of global solutions (see Martin and Pierre [10 ] and Hollis [4]). The purpose of this paper is to prove existence of a global solution using only one inequality in the case of 3 component systems. Our technique is based on the construction of polynomial functionals (according to solutions of the reaction-diffusion equations) which give, using the well known regularizing effect, the global existence. This result generalizes those obtained recently by Kouachi [6] and independently by Malham and Xin [9]. Submitted December 13, 2001. Published October 16, 2002. Math Subject Classifications: 35K45, 35K57. Key Words: Reaction diffusion systems; Lyapunov functionals; global existenc