Perturbation analysis in a free boundary problem arising in tumor growth model

TL;DR

This paper investigates the existence, multiplicity, and bifurcation of solutions for a free boundary problem modeling tumor growth, involving a discontinuous nonlinearity, and employs perturbation techniques to analyze solutions near radial configurations.

Contribution

It introduces new methods to analyze free boundary problems with discontinuous nonlinearities in tumor growth models, including existence, multiplicity, and bifurcation results, and applies perturbation techniques for non-radial solutions.

Findings

Existence of radial solutions with bifurcation diagrams

Multiplicity of solutions depending on parameters

Characterization of free boundaries near radial solutions

Abstract

We study the existence and multiplicity of solutions of the following free boundary problem $$ (P)\left\{ \begin{array}{rcll} \del u &=& \lam ( \eps +(1-\eps ) H(u-\mu))~ \hspace{3mm}&\text{in}~\Omega (t)\\ u&=& \overline{u}_{\infty}~\hspace{3mm} &\text{on } ~ \partial \Omega(t) \end{array} \right. $$ where $\Omega(t) \subset \RR^3$ a regular domain at $t>0$, $\eps,~\overline{u}_{\infty},~\lambda,~\mu$ are a positive parameters and $H$ is the Heaviside step function. \\The problem (P) has two free boundaries: the outer boundary of $\Omega(t)$ and the inner boundary whose evolution is implicit generated by the discontinuous nonlinearity $H$. The problem (P) arise in tumor growth models as well as in other contexts such as climatology. First, we show the existence and multiplicity of radial solutions of problem (P) where $\Omega(t)$ is a spherical domain. Moreover, the bifurcation diagrams are giving. Secondly, using the perturbation technic combining to local methods, we prove the existence of solutions and characterize the free boundaries of problem (P) near the corresponding radial solutions.