Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.09281 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917398676766720 |
|---|---|
| author | Gómez-Castro, David Płociniczak, Łukasz Vázquez, Juan Luis |
| author_facet | Gómez-Castro, David Płociniczak, Łukasz Vázquez, Juan Luis |
| contents | We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range $m > 1$ and positive solutions with heavy tails in the sub-critical fast-diffusion range $m_c < m < 1$. The self-similar solutions in the linear case $m=1$ were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit $m \to 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09281 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-similar solutions to the time-fractional Porous-Medium Equation Gómez-Castro, David Płociniczak, Łukasz Vázquez, Juan Luis Analysis of PDEs We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range $m > 1$ and positive solutions with heavy tails in the sub-critical fast-diffusion range $m_c < m < 1$. The self-similar solutions in the linear case $m=1$ were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit $m \to 1$. |
| title | Self-similar solutions to the time-fractional Porous-Medium Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.09281 |