This paper discusses the asymptotic stability of the equilibrium states of a nonlinear diffusion system with time delays. It is assumed that the system is describable by a partial differential-difference equation of the form:
$∂u(t,X)∂t=ℒu(t,X)+f(t,X,u(t,X),$
$u(t−T,X),…,∂u(t,X)∂xi,…,$
$∂u(t−T,X)∂xi,…)i=1,…,M$
where ℒ is a linear operator uniformly elliptic in X defined for all XεΩ … a bounded, open M-dimensional spatial domain; f is a specified function of its arguments. In the development of this paper, the physical origin of the foregoing equation is discussed briefly. Then, conditions for asymptotic stability of the trivial solution are derived via an extended Lyapunov’s direct method. Specific results are given for a simple one-dimensional linear heat equation with time-delayed arguments.
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