Well-posedness for a class of phase-field systems modeling prostate cancer growth with fractional operators and general nonlinearities

TL;DR

This paper establishes well-posedness for a complex phase-field model of prostate cancer growth involving fractional operators, nonlinearities, and couplings, extending previous models with rigorous mathematical analysis.

Contribution

It provides the first comprehensive existence, uniqueness, and continuous dependence results for a fractional, nonlinear prostate cancer model with general potentials.

Findings

Proved existence and uniqueness of solutions.

Handled nonsmooth potentials and general nonlinearities.

Extended analysis to fractional operators in the model.

Abstract

This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. 30 (2020), 1253-1295], preprint in arXiv:1907.11618 [math.AP]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a Allen-Cahn equation with double-well potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties.