Abstract
In this paper, we continue the research in Li et al. (J Differ Equ 371:353–395, 2023) and study the linear and global stability of a class of reaction-diffusion systems with general degenerate diffusion. The establishment of these systems is based on the acid-mediated invasion hypothesis, which is a candidate explanation for the Warburg effect. Our theoretical results characterize the effects of acid resistance and mutual competition between healthy cells and tumor cells on local and long-term tumor development, i.e., whether the healthy cells and tumor cells coexist or the tumor cells prevail after tumor invasion. We first consider the linear stability of the steady states and give a complete characterization by transforming the linearized analysis into an algebraic problem. In discussing global stability, the main difficulty of this model arises from density-limited diffusion terms, which can lead to degeneracy in the parabolic equations. We find that the method established in Li et al. (J Differ Equ 371:353–395, 2023) works well to overcome the degenerate problem. This method combines the Lyapunov functionals and upper/lower solutions, and it can be applied to a broader range of reaction-diffusion systems even if the diffusion terms degenerate and have very poor properties.
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Acknowledgements
The first author is supported by NSF of China (Nos. 12371213, 12126609) and R &D project of Pazhou Lab (Huangpu) (No. 2023K0601). The second author is supported by the National Key R &D Program of China (No. 2020YFA0712500), NSF of China (No. 12126609), R &D project of Pazhou Lab (Huangpu) (No. 2023K0601) and Shenzhen Science and Technology Program (No. CJGJZD20210408091403008).
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Li, F., Yao, Za. & Yu, R. A general degenerate reaction-diffusion model for acid-mediated tumor invasion. Z. Angew. Math. Phys. 75, 75 (2024). https://doi.org/10.1007/s00033-024-02220-z
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DOI: https://doi.org/10.1007/s00033-024-02220-z