An estimate of growth bound of positive C₀-semigroup on L^p space and its applications

Let {T(t)}_t≥0 be a positive C₀-semigroup on L^p(Ω), with infinitesimal generator A. In this paper, it is proved that if there exists a c ∈ L^∞(Ω) ∩ D(A*) such that ess inf _r∈Ω c(r) > 0 and b := ess sup _x∈Ω (A*c)(x)/c(x) < ∞ 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 < ∞ and c ∈ D(A), where a = ess sup _x∈Ω (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.