Convex semigroups on Lp-like spaces

Robert Denk, Michael Kupper and Max Nendel
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Robert Denk: Center for Mathematical Economics, Bielefeld University
Michael Kupper: Center for Mathematical Economics, Bielefeld University
Max Nendel: Center for Mathematical Economics, Bielefeld University

No 712, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

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.

Keywords: Convex semigroup; nonlinear Cauchy problem; well-posedness and uniqueness; Hamilton-Jacobi-Bellman equation (search for similar items in EconPapers)
Pages: 25
Date: 2025-06-24
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