Asymptotic estimates in Thompson’s metric for semilinear parabolic equations

In ordered Banach spaces we prove exponential asymptotic stability estimates in Thompson’s metric for the semilinear equation u′=Au+f(u)u′=Au+f(u), where AA is the generator of a positive and exponentially stable C0C0-semigroup and ff is a contraction with respect to Thompson’s metric. We apply our results to the study of the semilinear parabolic equation u′(t)=Δu(t)+u(t)κu′(t)=Δu(t)+u(t)κ with homogeneous Dirichlet boundary conditions on bounded domains where ∣κ∣<1∣κ∣<1.