On Stability of Vector Nonlinear Integrodifferential Equations

Let Ω be a bounded domain in a real Euclidean space. We consider the equation ∂u(t,x)/∂t=C(x)u(t,x)+∫ΩK(x,s)u(t,s)ds+[F(u)](t,x)  (t>0;  x∈Ω), where C(·) and K(·,·) are matrix-valued functions and F(·) is a nonlinear mapping. Conditions for the exponential stability of the steady state are established. Our approach is based on a norm estimate for operator commutators