On variable Lebesgue spaces and generalized nonlinear heat equations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915174294749184 |
|---|---|
| author | Vergara-Hermosilla, Gastón |
| author_facet | Vergara-Hermosilla, Gastón |
| contents | In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2α- \langle 1 \rangle_γ} } (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On variable Lebesgue spaces and generalized nonlinear heat equations Vergara-Hermosilla, Gastón Analysis of PDEs In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some structural properties of these spaces with decay estimates of the fractional heat kernel, we were able to prove two well-posedness results for these equations. In a first theorem, we prove the existence and uniqueness of global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{ \frac{nb}{2α- \langle 1 \rangle_γ} } (\mathbb{R}^n,L^\infty([0,T[ ))$. On the other hand, by introducing a new class of variable exponents, we demonstrate the existence of an unique local-in-time mild solution in the space $L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$. |
| title | On variable Lebesgue spaces and generalized nonlinear heat equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.09588 |