Asymptotic analysis of a tumor growth model with fractional operators

Abstract

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 Colli -Gilardi -Sprekels 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.

A Distributed Control Problem for a Fractional Tumor Growth Model

“…In this paper, we first establish similar results for system (1.1)-(1.3) by assuming α = 0 and β > 0 (in fact, we take β = 1 without loss of generality). In particular, we extend some results shown for this case in the recent paper [17]. Then, we discuss a distributed control problem for the modified system.…”

“…Remark 2.7. The existence part of Theorem 2.6 is closely connected to the existence result proved in [17,Theorem 3.4], where, however, no statement concerning separation or uniqueness was proved. For purposes of control theory, however, it is indispensable to have uniqueness, since otherwise no control-to-state operator can be defined, and this seems to be available only under the assumptions made here.…”

“…While the occurrence of more general diffusional regimes of fractional type has been studied for a long time in the mathematical literature, it was only recently (see, e.g., [1, 2, 12-14, 16, 28-30]) that fractional operators have been investigated in the framework of Cahn-Hilliard systems (for phase field systems of Caginalp type, see also [5]), and the only investigations of tumor growth models involving fractional diffusive regimes such as in the system (1.1)- (1.3) seem to be the recent papers [15,17] by the present authors.…”

Well-posedness for a class of phase-field systems modeling prostate cancer growth with fractional operators and general nonlinearities

“…In recent years the phase-field (or diffuse interface) method has been extensively used to describe tumor growth in the computational and mathematical literature (see, e.g., [8,9,11,13,15,26,32,34,37,39,40,44,46,47,49,[54][55][56]). Indeed, tumor growth has become an important issue in applied mathematics and a significant number of models has been introduced and discussed, with numerical simulations as well, in connection and comparison with the behavior of other special materials: one may also see [5,7,12,16,20,25,27,28,31,33,35,36,42,48,50,52,53].…”

“…For an overview of recent contributions, we refer to the papers [14,18] and [10], which offer to the interested reader a number of suggestions to expand the knowledge of the field. Moreover, we underline that the authors of the present paper already investigated systems with fractional operators in the papers [15][16][17][19][20][21][22], in particular studying another class of tumor growth models [15,20] inspired by [40] and the related contributions [11,13,34,50]. In our approach here, we adopt the same setting for fractional operators, that are defined through the spectral theory.…”