Convex semigroups on $$L^p$$ L p -like spaces

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.