Non-linear instability and exact solutions to some delay reaction–diffusion systems

Abstract

We deal with coupled delay non-linear reaction–diffusion systems of the formut=k1uxx+F(u,u¯,w,w¯),wt=k2wxx+G(u,u¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), and τ is the delay time. For a wide class of the kinetic functions F and G, we determine global instability conditions; once these conditions hold, any solution of the system is unstable. The solution instability is proved with an exact approach without making any assumptions or approximations (this approach can be useful for analyzing other non-linear delay models, including biological, biochemical, biophysical, etc.). We discuss some ill-posed Cauchy-type and initial-boundary-value problems connected with the global instability. We present multiparameter exact solutions involving an arbitrary number of free parameters and give an exact solution that represents a non-linear superposition of a traveling wave and a periodic standing wave. All of the systems considered contain two arbitrary functions of two or three arguments. We also study other non-linear systems of delay PDEs including reaction–diffusion system with two different time-varying delays, multicomponent systems of reaction–diffusion equations and more complex, higher-order non-linear systems with delay. The results may be used in solving certain problems and testing approximate analytical and numerical methods for certain classes of similar and more complex non-linear delay systems.