WIAS Preprint No. 1956, (2014)

On a diffuse interface model of tumor growth



Authors

  • Frigeri, Sergio Pietro
  • Grasselli, Maurizio
  • Rocca, Elisabetta

2010 Mathematics Subject Classification

  • 35D30 35K57 35Q92 37L30 92C17

Keywords

  • diffuse interface, tumor growth, Cahn-Hilliard equations, reaction-diffusion equations, weak solutions, well-posedness, global attractors

Abstract

We consider a diffuse interface model of tumor growth proposed by A. Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor 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 functionp(φ) multiplied by the differences of the chemical potentials for φ and ψ. The system is equipped with no-flux boundary conditions which entails 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.

Appeared in

  • European J. Appl. Math., 26 (2015) pp. 215--243.

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