Some conditions for global asymptotic stability of equilibria of integrodifferential equations (Q762409)

The present paper deals with some conditions for global asymptotic stability of equilibria of integrodifferential equations of the following form \[ (1)\quad \partial /\partial t\quad u(t,x)=D\Delta u(t,x)+f(u(t,x),(Ru)(t,x)),\quad (t,x)\in R_+\times \Omega, \] \(\partial /\partial \eta\) \(u(t,y)=0\), \((t,y)\in R_+\times \partial \Omega\), \(u(t,x)=h(t,x)\), \((t,x)\in R_-\times {\bar \Omega}\) where \(u(t,x)\in R^ n\) and \[ (Ru)_ i(t,x):=\int^{+\infty}_{0}d\eta_ i(s)(\int_{\Omega}G_ i(s,x,y)u_ i(t-s,y)\quad dy). \] In section 2 the author expresses the problem (1) in an abstract form and obtains some theorems on invariance and attractivity of convex sets. In section 3, he states the results for a single equation \((n=1)\). The proofs are given in section 4. The n-dimensional case \((n>1)\) is discussed in section 5. In section 6, he discusses some applications.