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In this study, we analyze the behavior of monotone traveling waves of aone-dimensional porous medium equation modeling mechanical properties of livingtissues. We are interested in the asymptotics where the pressure, which governsthe diffusion process and limits the creation of new cells, becomes very stiff,and the porous medium equation degenerates towards a free boundary problem ofHele-Shaw type. This is the so-called incompressible limit. The solutions ofthe limit Hele-Shaw problem then couple "free dynamics" with zero pressure, and"incompressible dynamics" with positive pressure and constant density. In thefirst part of the work, we provide a refined description of the traveling wavesfor the porous medium equation in the vicinity of the transition between thefree domain and the incompressible domain. The second part of the study isdevoted to the analysis of the stability of the traveling waves. We prove thatthe 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 $L^\infty$ controlstemming from the maximum principle. As a consequence, we prove that travelingwaves are stable under small perturbations.

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