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We consider a phase-field model of prostate cancer growth with chemotherapyand antiangiogenic therapy effects which is introduced in [2]. It is comprisedof phase-field equation to describe tumor growth, which is coupled to areaction-diffusion type equation for generic nutrient for the tumor. Anadditional equation couples the concentration of prostate-specific antigen(PSA) in the prostatic tissue and it obeys a linear reaction-diffusionequation. The system completes with homogeneous Dirichlet boundary conditionsfor the tumor variable and Neuman boundary condition for the nutrient and theconcentration of PSA. Here we investigate the long time dynamics of the model.We first prove that the initial-boundary value problem generates a stronglycontinuous semigroup on a suitable phase space that admits the global attractorin a proper phase space. Moreover, we also discuss the convergence of asolution to a single stationary state and obtain a convergence rate estimateunder some conditions on the coefficients.

Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects