Behavior of solutions and chaos in nonlinear reaction-diffusion PDE's related to cancer

TL;DR

This paper investigates the solutions and chaotic behavior of nonlinear reaction-diffusion PDEs with nonlocal nonlinearities and variable exponents, revealing conditions for chaos and implications for cancer dynamics.

Contribution

It introduces new analysis of nonlinear PDEs with nonlocal terms, showing how solutions can become non-dissipative and generate complex chaos relevant to cancer modeling.

Findings

Existence of infinite unstable solutions with varying speeds.

Generation of spatio-temporal chaos through bifurcations.

Behavior of solutions linked to cancer dynamics.

Abstract

In this paper, we study the mixed problem for new class of nonlinear reaction-diffusion PDEs with the nonlocal nonlinearity with variable exponents. Here we obtain results on solvability and behavior of solutions both when these are yet dissipative and when these get become non-dissipative problems. These problems are possess special properties: these can be to remain dissipative all time or can get become non dissipative after finite time. It is shown that if the studied problems get become non-dissipative can have an infinite number of different unstable solutions with varying speeds and also an infinite number of different states of spatio-temporal (diffusion) chaos that are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and it is explained how space-time chaos can arise. Since these problems desctibe the dynamics of cancer, we explain the behavior of the dynamics of cancer using obtained here results.