Null-controllability of perturbed porous medium gas flow^*

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain.
² Chair of Computational Mathematics, Fundación Deusto, Av. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain.

^** Corresponding author: borjan.geshkovski@uam.es

Received: 6 September 2019
Accepted: 27 February 2020

Abstract

In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.