Advances in Mathematical
Sciences and Applications
Vol. 28, No. 2 (2019), pp. 343–375

GAKKOTOSHOTOKYO JAPANWELL-POSEDNESS AND REGULARITY
FOR A FRACTIONAL TUMOR GROWTH MODEL

Pierluigi Colli
Dipartimento di Matematica “F. Casorati”, Università di Pavia
and Research Associate at the IMATI – C.N.R. Pavia
via Ferrata 5, 27100 Pavia, Italy(E-mail: pierluigi.colli@unipv.it)

Gianni Gilardi
Dipartimento di Matematica “F. Casorati”, Università di Pavia
and Research Associate at the IMATI – C.N.R. Pavia
via Ferrata 5, 27100 Pavia, Italy(E-mail: gianni.gilardi@unipv.it)

andJürgen Sprekels
Department of Mathematics, Humboldt-Universität zu Berlin
Unter den Linden 6, 10099 Berlin, Germany
and Weierstrass Institute for Applied Analysis and Stochastics
Mohrenstrasse 39, 10117 Berlin, Germany(E-mail: sprekels@wias-berlin.de)

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 generalization of a phase field system ofCahn–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
Communicated by Editors; Received June 26, 2019.
AMS Subject Classification: 35Q92, 92C17, 35K35, 35K90.
Keywords: fractional operators, Cahn–Hilliard systems, well-posedness, regularity of solutions, tumor
growth models.

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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 ismotivated by the possibility that the diﬀusional regimes governing the evolution of thediﬀerent 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, bywriting the equation governing the evolution of the chemical potential in the form of ageneral variational inequality, also singular or nonsmooth contributions of logarithmic orof double obstacle type to the energy density can be admitted.

1Introduction

Let Ω ⊂ R3 denote an open, bounded, and connected set with smooth boundary Γ andunit outward normal n, let T > 0 be given, and set Qt := Ω × (0, t) for t ∈ (0, T ) andQ := Ω × (0, T ), as well as Σ := Γ × (0, t). We investigate in this paper the evolutionary
systemα ∂t µ + ∂t φ + A2ρ µ = P (φ)(S − µ) in Q,µ = β ∂t φ + B 2σ φ + f (φ) in Q,∂t S + C 2τ S = −P (φ)(S − µ) in Q,µ(0) = µ0 , φ(0) = φ0 , S(0) = S0 , in Ω.

(1.1)
(1.2)
(1.3)
(1.4)

In the above system, α > 0 and β > 0, and A2ρ , B 2σ , C 2τ , with r, σ, τ > 0, denote fractional powers of the selfadjoint, monotone, and unbounded linear operators A, B, and C,
respectively, which are supposed to be densely defined in H := L2 (Ω) and to have compact resolvents. Moreover, f denotes the derivative of a double-well potential F . Typicaland physically significant examples of F are the so-called classical regular potential, thelogarithmic potential , and the double obstacle potential , which are given, in this order, by
1 2(r − 1)2 , r ∈ R,4 ()r ∈ (−1, 1)
 (1 + r) ln(1 + r) + (1 − r) ln(1 − r) − c1 r2 ,2 log(2) − c1 ,r ∈ {−1, 1} ,Flog (r) :=+∞ ,r ̸∈ [−1, 1])(if |r| ≤ 1 and F2obs (r) := +∞ if |r| > 1.
F2obs (r) := c2 1 − r2Freg (r) :=

(1.5)
(1.6)
(1.7)

Here, the constants ci in (1.6) and (1.7) satisfy c1 > 1 and c2 > 0, so that the corresponding
functions are nonconvex. In cases like (1.7), one has to split F into a nondiﬀerentiable
convex part F1 (the indicator function of [−1, 1], in the present example) and a smooth
perturbation F2 . Accordingly, in the term f (φ) appearing in (1.2), one has to replace thederivative F1′ of the convex part F1 by the subdiﬀerential f1 := ∂F1 and interpret (1.2) as adiﬀerential inclusion or as a variation inequality involving F1 rather than f1 . Furthermore,
the function P occurring in (1.1) and (1.3) is nonnegative and smooth. Finally, the termson the right-hand sides in (1.4) are prescribed initial data.