Behavior of Solutions Nonlinear Reaction-Diffusion PDE's Relation to Dynamics of Propagation of Cancer

TL;DR

This paper introduces a new nonlinear reaction-diffusion PDE model to describe cancer propagation, analyzing solution behaviors including dissipation, chaos, and instability, providing insights into the complex dynamics of cancer spread.

Contribution

The paper develops a novel PDE model with nonlocal nonlinearities and variable exponents, analyzing its solutions' behavior and their implications for cancer propagation dynamics.

Findings

Solutions can be dissipative or non-dissipative, with transitions between these states.

Chaotic and unstable solutions with varying propagation speeds can occur.

The model explains complex phenomena like space-time chaos in cancer spread.

Abstract

In this paper, we propose a new mathematical model nonlinear reaction-diffusion PDE's describing the dynamics of propagation of cancer. Here the mixed problem for the proposed PDE's is investigated and by applying obtained results conclusions on the dynamics of propagation of cancer are drawn. These problems have nonlocal nonlinearity with variable exponents and possess special properties: these can be to remain either dissipative all time or become non-dissipative after a finite time. Here the solvability and behavior of solutions both when problems are yet dissipative and when become nondissipative are proved. It is shown that if the studied process gets become nondissipative can have various states, e.g. an infinite number of different unstable solutions with varying speeds, in addition, their propagation can become chaotic. The behavior of these solutions is analyzed in detail and it is explained how space-time chaos can arise. Investigation of this mathematics model allows explaining the dynamics of propagation of cancer, which are provided here as conclusions for each case.