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On a diffuse interface model of tumour growth

Published online by Cambridge University Press:  20 January 2015

SERGIO FRIGERI ,
MAURIZIO GRASSELLI  and
ELISABETTA ROCCA
SERGIO FRIGERI
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany email: SergioPietro.Frigeri@wias-berlin.de
MAURIZIO GRASSELLI
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Milano I-20133, Italy email: maurizio.grasselli@polimi.it
ELISABETTA ROCCA
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany email: Elisabetta.Rocca@wias-berlin.de Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Milano I-20133, Italy email: elisabetta.rocca@unimi.it
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Abstract

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

Keywords

diffuse interfacetumour growthCahn–Hilliard equationsreaction–diffusion equationsweak solutionswell-posednessglobal attractors

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Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 215 - 243
Copyright
Copyright © Cambridge University Press 2015 

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