Asynchronous exponential growth of semigroups of nonlinear operators (Q1197411)

The authors analyze the property of asynchronous exponential growth for the abstract nonlinear differential equation \(z'(t)= Az(t)+ F(z(t))\), \(t\geq 0\), \(z(0)= x\in X\), where \(A\) is the infinitesimal generator of a semigroup of linear operators in the Banach space \(X\) and \(F\) is a nonlinear operator in \(X\). Asynchronous exponential growth means that the nonlinear semigroup \(S(t)\), \(t\geq 0\) associated with this problem has the property that there exists \(\lambda> 0\) and a nonlinear operator \(Q\) in \(X\) such that the range of \(Q\) lies in a one-dimensional subspace of \(X\) and \(\lim_{t\to \infty} e^{-\lambda t} S(t) x= Qx\) for all \(x\in X\). It is proved that if the linear semigroup generated by \(A\) has asynchronous exponential growth and \(F\) satisfies \(\| F(x)\|\leq f(\| x\|)\| x\|\), where \(f: \mathbb{R}_ +\to \mathbb{R}_ +\) and \(\int^ \infty (f(r)/r)dr< \infty\), then the nonlinear semigroup \(S(t)\), \(t\geq 0\) has asynchronous exponential growth. The method of proof is a linearization about infinity. Examples from structured population dynamics are given to illustrate the results.