Maximal regularity and optimal control for a non-local Cahn-Hilliard tumour growth model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913446290784256 |
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| author | Fornoni, Matteo |
| author_facet | Fornoni, Matteo |
| contents | We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted $L^p$ spaces. This technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal distributed control problem, aimed at identifying a suitable therapy guiding the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal regularity and optimal control for a non-local Cahn-Hilliard tumour growth model Fornoni, Matteo Analysis of PDEs Optimization and Control 35K61, 45K05, 35B65, 35Q92, 49K20, 92C50 We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. First, we establish novel regularity results for such a model, by applying maximal regularity theory in weighted $L^p$ spaces. This technique enables us to prove the local existence and uniqueness of a regular solution, including also chemotaxis effects. By leveraging time-regularisation properties and global boundedness estimates, we further extend the solution to a global one. These results provide the foundation for addressing an optimal distributed control problem, aimed at identifying a suitable therapy guiding the tumour towards a predefined target. Specifically, we prove the existence of an optimal therapy and, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. |
| title | Maximal regularity and optimal control for a non-local Cahn-Hilliard tumour growth model |
| topic | Analysis of PDEs Optimization and Control 35K61, 45K05, 35B65, 35Q92, 49K20, 92C50 |
| url | https://arxiv.org/abs/2402.04204 |