Asymptotic stability of constant steady states for a 2×2 reaction–diffusion system arising in cancer modelling
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AbstractThe dependence of tumor on essential nutrients is known to be crucial for its evolution and has become one of the targets for medical therapies. Based on this fact a reaction-diffusion system with chemotaxis term and nutrient-based growth of tumors is presented. The formulation of the model considers also an influence of tumor and pharmacological factors on nutrient concentration. In the paper, convergence of solutions to constant, stationary states in the one-dimensional case for small perturbation of the equilibria is investigated. The nonlinear stability results are obtained by means of the classical symmetrization method and energy Sobolev estimates. © 2010 Elsevier Ltd.
CitationDi Francesco M, Twarogowska M (2011) Asymptotic stability of constant steady states for a 2×2 reaction–diffusion system arising in cancer modelling. Mathematical and Computer Modelling 53: 1457–1468. Available: http://dx.doi.org/10.1016/j.mcm.2010.03.034.
SponsorsMDF acknowledges support from the KAUST award of Prof. Peter A. Markowich (University of Cambridge). The authors are grateful to Prof. Luigi Preziosi for helpful suggestions and comments. Part of this work was performed when the authors were attending an advanced course in Biomathematics at CRM (Autonomous University of Barcelona): they are grateful to the organizers for their support. Finally, the authors would like to thank the two referees for their useful suggestions on the improvement of the paper.