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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.08768 |
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| _version_ | 1866915488678805504 |
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| author | He, Qingyou |
| author_facet | He, Qingyou |
| contents | The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on their respective supports for the two free boundary problems. For the pressure of the porous medium type reaction-diffusion equation, the $\frac{1}{2}$-concavity preserves all the time, while $α$-concavity for $α\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ does not persist in time. In contrast, in the case of the pressure for the Hele-Shaw problem, $α$-concavity with $α\in[0,\frac{1}{2}]$ is maintained all the while and $\frac{1}{2}$ acts as the largest index. The intuitive explanation for the difference between the two free boundary problems is that, although the Hele-Shaw problem is the incompressible limit of the porous medium-type reaction-diffusion equation, it is no longer a degenerate parabolic equation. Furthermore, for the pressure of the porous medium type reaction-diffusion equation, the non-degenerate estimate is established by means of the derived concave properties, indicating that the spatial Lipschitz regularity in the whole space is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp power concavity of two relevant free boundary problems of reaction-diffusion type He, Qingyou Analysis of PDEs The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on their respective supports for the two free boundary problems. For the pressure of the porous medium type reaction-diffusion equation, the $\frac{1}{2}$-concavity preserves all the time, while $α$-concavity for $α\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ does not persist in time. In contrast, in the case of the pressure for the Hele-Shaw problem, $α$-concavity with $α\in[0,\frac{1}{2}]$ is maintained all the while and $\frac{1}{2}$ acts as the largest index. The intuitive explanation for the difference between the two free boundary problems is that, although the Hele-Shaw problem is the incompressible limit of the porous medium-type reaction-diffusion equation, it is no longer a degenerate parabolic equation. Furthermore, for the pressure of the porous medium type reaction-diffusion equation, the non-degenerate estimate is established by means of the derived concave properties, indicating that the spatial Lipschitz regularity in the whole space is sharp. |
| title | Sharp power concavity of two relevant free boundary problems of reaction-diffusion type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.08768 |