2025/08/27 by He, Wenhua, Wang, Mingxin, Xing, Ruixiang
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.19954
In this paper, we consider a free boundary multi-layer tumor model that incorporates a T-periodic provision of external nutrients Φ(t). The simplified model contains three parameters: the mean of periodic external nutrients Φ(t), the threshold concentration \widetildeσ for proliferation and the cell to cell adhesiveness coefficient γ. We first study the flat solution and give a complete classification about (1)/(T) ∫0T Φ(t) d t and \widetildeσ according to global stability of zero equilibrium solution or global stability of the positive periodic solution. Precisely, (i) a zero flat solution is globally stable under the flat perturbations if and only if \widetildeσ \geqslant (1)/(T) ∫0T Φ(t) d t; (ii) If \widetildeσ<(1)/(T) ∫0T Φ(t) d t, then there exists a unique positive flat solution (σ_*(y, t), p_*(y, t), ρ_*(t)) with period T and it is a global attractor of all positive flat solutions for all γ>0. We further investigate periodic solutions bifurcating from the flat periodic solution (σ_*(y, t), p_*(y, t), ρ_*(t)). By periodicity and symmetry, we not only give symmetry-breaking periodic solutions for all positive parameter γj, but also show the existence of a plethora of periodic bifurcations. For the free boundary tumor problem, this is the first result of the existence of periodic bifurcations.