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Asymptotic analysis of a tumor growth model with fractional operators
- Source :
- Asymptotic Analysis. 120:41-72
- Publication Year :
- 2020
- Publisher :
- IOS Press, 2020.
-
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, 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.<br />Comment: Key words: fractional operators, Cahn-Hilliard systems, well-posedness, regularity of solutions, tumor growth models, asymptotic analysis
- Subjects :
- 35K90
Asymptotic analysis
Generalization
General Mathematics
35Q92
Type (model theory)
01 natural sciences
Mathematics - Analysis of PDEs
Operator (computer programming)
Fractional operators
well-posedness
FOS: Mathematics
Applied mathematics
0101 mathematics
Mathematics
regularity of solutions
35B40
010102 general mathematics
Relaxation (iterative method)
Function (mathematics)
35B40, 35K55, 35K90, 35Q92, 92C17
92C17
010101 applied mathematics
asymptotic analysis
Monotone polygon
Cahn--Hilliard systems
35K55
Variational inequality
tumor growth models
Analysis of PDEs (math.AP)
Subjects
Details
- ISSN :
- 18758576 and 09217134
- Volume :
- 120
- Database :
- OpenAIRE
- Journal :
- Asymptotic Analysis
- Accession number :
- edsair.doi.dedup.....1732516034317f6ecf4c98dd79bbdbe2