An Estimate of Growth Bound of Positive C0-Semigroup on Lp Space and its Applications

Let {T(t)}t => 0 be a positive C0-semigroup on Lp(omega), with infinitesimal generator A. In this paper, it is proved that if there exists a c (element of) L^(infinity) (Omega) (intersection) D(A*) such that ess inf c(r) > 0 and b : ess sup (A*(c))(x)/c(x) < (infinity) , where A* is the adjoint of A, then the growth bound of T(t) is upper bounded by b when p = 1, and by b/p + a/q when 1 < p < (inifinity) and c (element of) D(A), where a = ess sup (x (element of) Omega) (Ac)(x)/c(x) . This is an operator version of a classical stability result on Z-matrix. As application examples, some new results on the asymptotic behaviours of population system and neutron transport system are obtained.