# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021140
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## Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects

 1 Dipartimento di Matematica, Universita' degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy 2 Dipartimento di Matematica, Universita' degli Studi di Pavia and IMATI-C.N.R., Via Ferrata 5, 27100 Pavia, Italy

* Corresponding author

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: Tania Biswas would like to thank Department of Mathematics, University of Pavia for providing financial support and stimulating environment for the research and Prof. Elisabetta Rocca, Prof. Pierluigi Colli for fruitful discussions. This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics "F. Casorati", University of Pavia. In addition, this research has been performed in the framework of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR "Matematica d'Eccellenza in biologia ed ingegneria come acceleratore di una nuova strateGia per l'ATtRattività dell'ateneo pavese". The present paper also benefits from the support of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica)

We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [2]. It is comprised of phase-field equation to describe tumor growth, which is coupled to a reaction-diffusion type equation for generic nutrient for the tumor. An additional equation couples the concentration of prostate-specific antigen (PSA) in the prostatic tissue and it obeys a linear reaction-diffusion equation. The system completes with homogeneous Dirichlet boundary conditions for the tumor variable and Neumann boundary condition for the nutrient and the concentration of PSA. Here we investigate the long time dynamics of the model. We first prove that the initial-boundary value problem generates a strongly continuous semigroup on a suitable phase space that admits the global attractor in a proper phase space. Moreover, we also discuss the convergence of a solution to a single stationary state and obtain a convergence rate estimate under some conditions on the coefficients.

Citation: Tania Biswas, Elisabetta Rocca. Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021140
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