Asymptotic estimates in Thompson's metric for semilinear parabolic equations. (January 2016) - Herzog, Gerd; Kunstmann, Peer Chr.
Abstract: In ordered Banach spaces we prove exponential asymptotic stability estimates in Thompson's metric for the semilinear equation u ′ = A u + f ( u ), where A is the generator of a positive and exponentially stable C 0 -semigroup and f is a contraction with respect to Thompson's metric. We apply our results to the study of the semilinear parabolic equation u ′ ( t ) = Δ u ( t ) + u ( t ) κ with homogeneous Dirichlet boundary conditions on bounded domains where ∣ κ ∣ < 1 .