Dynamics of Partial Differential Equations

Volume 15 (2018)

Number 2

On the vanishing viscosity approximation of a nonlinear model for tumor growth

Pages: 95 – 108

DOI: https://dx.doi.org/10.4310/DPDE.2018.v15.n2.a1


Donatella Donatelli (Department of Engineering Computer Science and Mathematics, University of L’Aquila, Italy)

Konstantina Trivisa (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)


We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Darcy’s equation. The domain occupied by the tumor in this setting is a growing continuum $\Omega$ with boundary $\partial \Omega$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on the vanishing viscosity of the Brinkman’s regularization. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data.


tumor growth models, cancer progression, mixed models, moving domain, penalization, existence

2010 Mathematics Subject Classification

Primary 35Q30, 76N10. Secondary 46E35.

The work of D.D. was supported by the Ministry of Education, University and Research (MIUR), Italy under the grant PRIN 2012- Project N. 2012L5WXHJ, Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: theoretical and applicative aspects. K.T. gratefully acknowledges the support in part by the National Science Foundation under the grant DMS-1211519 and by the Simons Foundation under the Simons Fellows in Mathematics Award 267399.

Received 12 October 2015

Published 29 May 2018