Fujita exponent for heat equation with Hörmander vector fields
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914140542468096 |
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| author | Chatzakou, Marianna Kassymov, Aidyn Ruzhansky, Michael |
| author_facet | Chatzakou, Marianna Kassymov, Aidyn Ruzhansky, Michael |
| contents | In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying Hörmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $φ(t)f(u)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fujita exponent for heat equation with Hörmander vector fields Chatzakou, Marianna Kassymov, Aidyn Ruzhansky, Michael Analysis of PDEs In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying Hörmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $φ(t)f(u)$. |
| title | Fujita exponent for heat equation with Hörmander vector fields |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.04196 |