Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid Growth
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908666177781760 |
|---|---|
| author | Fujishima, Yohei Hisa, Kotaro Laister, Robert |
| author_facet | Fujishima, Yohei Hisa, Kotaro Laister, Robert |
| contents | We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical $L^q$-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into $L^{\infty}$. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06104 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid Growth Fujishima, Yohei Hisa, Kotaro Laister, Robert Analysis of PDEs We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical $L^q$-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into $L^{\infty}$. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness. |
| title | Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid Growth |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.06104 |