Core Concepts
The core message of this article is to establish Hölder continuous dependence of the solutions of the porous medium equation with a growth term on the diffusion exponent.
Abstract
The article considers a macroscopic model for the growth of living tissues that incorporates 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.
The key highlights and insights are:
The authors prove Hölder continuous dependence of the solutions of the model on the diffusion exponent. This is an important result for validating and calibrating the model using experimental data.
The main difficulty lies in the degeneracy of the porous medium equations at vacuum. To address this, the authors first regularize the equation by shifting the initial data away from zero and then optimize the stability estimate derived in the regular setting.
The authors discuss the importance of this stability result for applying inverse problem methodologies, both in deterministic approaches involving parameter estimation and in stochastic Bayesian approaches for model calibration.
The authors point out that without the stability of solutions with respect to parameters, solving the inverse problem becomes unfeasible, as both stochastic and deterministic approaches rely on numerical approximation, which involves addressing finite-dimensional problems.
The authors also discuss how the stability of the solution with respect to the diffusion exponent is required to design efficient Monte Carlo algorithms for Bayesian inference.