Global asymptotic stability for semilinear equations via Thompson's metric

In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u′ = Au + ꭍ(u), if A is the generator of a positive and exponentially stable C₀-semigroup and ꭍ is a contraction with respect to Thompson's metric. The given estimates show that convergence holds with a uniform exponential rate.peerReviewe

Maximal L p -regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains ?, both with the following two domains of definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) : Bu = 0}, where B is the boundary operator.We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on Lp(?) which implies maximal regularity for the corresponding Cauchy problems. In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain D2(?) has the maximal regularity property, as in the case of smooth domains. In the last part,we construct an example that proves that, in general, the Dirichlet–Laplacian with domain D1(?) is not even a closed operator

Global Asymptotic Stability for Semilinear Equations via Thompson’s Metric

In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u ′ = Au + f(u), if A is the generator of a positive and exponentially stable C0-semigroup and f is a contraction with respect to Thompson’s metric. The given estimates show that convergence holds with a uniform exponential rate

Global asymptotic stability for semilinear equations via Thompson's metric

In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u′ = Au + ꭍ(u), if A is the generator of a positive and exponentially stable C₀-semigroup and ꭍ is a contraction with respect to Thompson's metric. The given estimates show that convergence holds with a uniform exponential rate.peerReviewe

Contraction semigroups on L∞(R)

If X is a non-degenerate vector field on R and H = −X 2 we examine conditions for the closure of H to generate a continuous semigroup on L ∞ which extends to the Lp-spaces. We give an example which cannot be extended and an example which extends but for which the real part of the generator on L2 is not lower semibounded