Convex semigroups on $L^p$-like spaces

TL;DR

This paper extends linear $C_0$-semigroup theory to convex semigroups on Banach lattices, establishing fundamental properties and conditions for well-posedness and continuity, with applications to nonlinear PDEs.

Contribution

It generalizes classical semigroup results to convex semigroups on Banach lattices, including generator properties and existence conditions for semigroup envelopes.

Findings

Generators of convex $C_0$-semigroups are closed and uniquely determine the semigroup.

The domain of the generator is invariant under the semigroup.

Conditions for the existence and strong continuity of semigroup envelopes are provided.

Abstract

In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $L^p$-spaces in mind as a typical application. We show that the basic results from linear $C_0$-semigroup theory extend to the convex case. We prove that the generator of a convex $C_0$-semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup; a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $C_0$-semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.