Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability of solutions of the porous medium equation with growth with respect to the diffusion exponent

Tomasz Dębiec, Piotr Gwiazda, Błażej Miasojedow, Zuzanna Szymańska

Abstract

We consider a macroscopic model for the growth of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Assuming a power-law relation between the mechanical pressure and the cell density, the model can be expressed as the porous medium equation with a growth term. We prove Hölder continuous dependence of the solutions of the model on the diffusion exponent. The main difficulty lies in the degeneracy of the porous medium equations at vacuum. To deal with this issue, we first regularise the equation by shifting the initial data away from zero and then optimise the stability estimate derived in the regular setting.

Stability of solutions of the porous medium equation with growth with respect to the diffusion exponent

Abstract

We consider a macroscopic model for the growth of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Assuming a power-law relation between the mechanical pressure and the cell density, the model can be expressed as the porous medium equation with a growth term. We prove Hölder continuous dependence of the solutions of the model on the diffusion exponent. The main difficulty lies in the degeneracy of the porous medium equations at vacuum. To deal with this issue, we first regularise the equation by shifting the initial data away from zero and then optimise the stability estimate derived in the regular setting.
Paper Structure (7 sections, 8 theorems, 77 equations)

This paper contains 7 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main result
  3. Basic properties of the solution
  4. Regularisation
  5. Stability estimate for regularised densities.
  6. Proof of the main theorem
  7. Conclusions and perspectives

Key Result

Theorem 2.1

Let $1< \gamma_{\mathrm{min}} < \gamma_{\mathrm{max}} < \infty$ be given and set $\Gamma=[\gamma_{\mathrm{min}},\gamma_{\mathrm{max}}]$. For $\gamma_1, \gamma_2 \in \Gamma$, let $u_{\gamma_1}$ and $u_{\gamma_2}$ be two solutions to nonlinear_diffusion with initial data $u_{0,\gamma_1}$ and $u_{0,\ga

Theorems & Definitions (17)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1: Nonnegativity of the density
  • proof
  • Proposition 3.2: Compact support property
  • proof
  • Proposition 3.3: $L^1$-contraction estimate
  • proof
  • Proposition 4.1: $L^\infty$ bounds and strict positivity of the regularised density
  • proof
  • ...and 7 more