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Author: Sprekels, Jürgen × DDC: 510 × Type: Text × Type: report × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Fractional operators × Subject: well-posedness ×

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    A distributed control problem for a fractional tumor growth model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24.) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work Adv. Math. Sci. Appl. 28 (2019), 343--375 by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.
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    Well-posedness and regularity for a fractional tumor growth model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn--Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. The generalization investigated in this paper is motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type. Under rather general assumptions, well-posedness and regularity results are shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth constributions of logarithmic or of double obstacle type to the energy density can be admitted.
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    Asymptotic analysis of a tumor growth model with fractional operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.
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    Well-posedness and regularity for a generalized fractional Cahn-Hilliard system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we investigate a rather general system of two operator equations that has the structure of a viscous or nonviscous Cahn–Hilliard system in which nonlinearities of double-well type occur. Standard cases like regular or logarithmic potentials, as well as non-differentiable potentials involving indicator functions, are admitted. The operators appearing in the system equations are fractional versions of general linear operators A and B, where the latter are densely defined, unbounded, self-adjoint and monotone in a Hilbert space of functions defined in a smooth domain and have compact resolvents. In this connection, we remark the fact that our definition of the fractional power of operators uses the approach via spectral theory. Typical cases are given by standard second-order elliptic differential operators (e.g., the Laplacian) with zero Dirichlet or Neumann boundary conditions, but also other cases like fourth-order systems or systems involving the Stokes operator are covered by the theory. We derive in this paper general wellposedness and regularity results that extend corresponding results which are known for either the non-fractional Laplacian with zero Neumann boundary condition or the fractional Laplacian with zero Dirichlet condition. These results are entirely new if at least one of the operators A and B differs from the Laplacian. It turns out that the first eigenvalue 1 of A plays an important und not entirely obvious role: if 1 is positive, then the operators A and B may be completely unrelated; if, however, 1 equals zero, then it must be simple and the corresponding one-dimensional eigenspace has to consist of the constant functions and to be a subset of the domain of definition of a certain fractional power of B. We are able to show general existence, uniqueness, and regularity results for both these cases, as well as for both the viscous and the nonviscous system.