Traveling waves for the porous medium equation in the incompressible limit: asymptotic behavior and nonlinear stability

In this study, we analyze the behavior of monotone traveling waves of a one-dimensional porous medium equation modeling mechanical properties of living tissues. We are interested in the asymptotics where the pressure, which governs the diffusion process and limits the creation of new cells, becomes very stiff, and the porous medium equation degenerates towards a free boundary problem of Hele-Shaw type. This is the so-called incompressible limit. The solutions of the limit Hele-Shaw problem then couple "free dynamics" with zero pressure, and "incompressible dynamics" with positive pressure and constant density. In the first part of the work, we provide a refined description of the traveling waves for the porous medium equation in the vicinity of the transition between the free domain and the incompressible domain. The second part of the study is devoted to the analysis of the stability of the traveling waves. We prove that the linearized system enjoys a spectral gap property in suitable weighted $L^2$ spaces, and we give quantitative estimates on the rate of decay of solutions. The nonlinear terms are treated perturbatively, using an $L^\infty$ control stemming from the maximum principle. As a consequence, we prove that traveling waves are stable under small perturbations.