Convex semigroups on $$L^p$$-like spaces
| Autor: | Robert Denk, Michael Kupper, Max Nendel |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Well-posedness and 01 natural sciences Domain (mathematical analysis) 010104 statistics & probability Mathematics - Analysis of PDEs Mathematics (miscellaneous) FOS: Mathematics Order (group theory) ddc:510 0101 mathematics Invariant (mathematics) Mathematics - Optimization and Control Mathematics 47H20 35A02 35A09 Cauchy problem Mathematics::Operator Algebras Semigroup Nonlinear Cauchy problem Probability (math.PR) 010102 general mathematics Regular polygon uniqueness Nonlinear system Optimization and Control (math.OC) Norm (mathematics) Convex semigroup Hamilton-Jacobi-Bellman equation Mathematics - Probability Analysis of PDEs (math.AP) |
| Zdroj: | Journal of Evolution Equations. 21:2491-2521 |
| ISSN: | 1424-3202 1424-3199 |
| DOI: | 10.1007/s00028-021-00693-3 |
| Popis: | 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. The manuscript has been split into two parts. The second part of the paper can be found under arXiv:2010.04594. 24 pages |
| Databáze: | OpenAIRE |
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