Nonexistence of solutions of certain semilinear heat equations
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910765196247040 |
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| author | Suragan, Durvudkhan Talwar, Bharat |
| author_facet | Suragan, Durvudkhan Talwar, Bharat |
| contents | We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11307 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonexistence of solutions of certain semilinear heat equations Suragan, Durvudkhan Talwar, Bharat Analysis of PDEs 35B33, 35B44 We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases. |
| title | Nonexistence of solutions of certain semilinear heat equations |
| topic | Analysis of PDEs 35B33, 35B44 |
| url | https://arxiv.org/abs/2210.11307 |