Non-linear instability and exact solutions to some delay reaction–diffusion systems (original) (raw)

Section snippets

Preliminary remarks. Single non-linear delay reaction–diffusion equations

Non-linear delay reaction–diffusion equations and coupled systems of such equations arise in biology, biophysics, biochemistry, chemistry, medicine, control, climate model theory, ecology, economics, and many other areas (e.g., see the studies [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and references in them). Similar equations occur in the mathematical theory of artificial neural networks and the results are used for signal and image processing as well as in image recognition problems

Exact method for proving instability of solutions to non-linear systems

Below we present the general idea of the method that we use to prove solution instability. Let a vector function u=u(x,t) be described by a non-linear system of partial differential equations (with or without delay) and let u0=u0(x,t) be a solution to this system. Suppose that one has managed to find another solution to the system in the form u=u0(x,t)+v(x,t,ε),where v=v(x,t,ε) is a sufficiently smooth function in all its arguments which is bounded everywhere at finite t and dependent on a

The class of non-linear delay reaction–diffusion systems

Section 3 will be dealing with a special class of non-linear reaction–diffusion systems of two coupled delay equations (4), (5) of the formut=k1uxx+bu+F(u−au¯,w,w¯),wt=k2wxx+G(u−au¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), with τ denoting the delay time. The kinetic functions F=F(ζ,w,w¯) and G=G(ζ,w,w¯) in Eqs. (7), (8) are arbitrary functions of three arguments, with F assumed to be non-degenerate in its first argument. For some other non-linear systems of delay PDEs, see

Multicomponent non-linear reaction–diffusion systems

Consider the following multicomponent reaction–diffusion system of delay equations:ut=kuxx+bu+F(x,t,u−au¯,w1,w¯1,…,wm,w¯m),(wn)t=kn(wn)xx+Gn(x,t,u−au¯,w1,w¯1,…,wm,w¯m),n=1,…,m,where u=u(x,t), u¯=u(x,t−τ), wn=wn(x,t), and w¯n=wn(x,t−τn), F(⋯) and Gn(⋯) are arbitrary functions of their arguments, and τ and τ n are delay times.

System (32) generalizes system (7), (8) in three different ways:

(i)
the number of equations involved can be arbitrary,
(ii)
the kinetic functions F and G n can additionally be dependent

Brief conclusions

To summarize, we have considered a wide class of non-linear reaction–diffusion systems of delay equationsut=k1uxx+bu+F(u−au¯,w,w¯),wt=k2wxx+G(u−au¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), F and G are arbitrary functions of three arguments, and τ is the delay time. We have proved that if the inequalities (global instability conditions) a>1,b>0,τ≥τ0,τ0=(lna)/bhold, any solution of the above system will be unstable for any kinetic functions F(ζ,w,w¯) and G(ζ,w,w¯). The global