A semilinear problem associated to the space-time fractional heat equation in $\mathbb{R}^N$
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913368208572416 |
|---|---|
| author | Cortázar, Carmen Quirós, Fernando Wolanski, Noemí |
| author_facet | Cortázar, Carmen Quirós, Fernando Wolanski, Noemí |
| contents | We study the fully nonlocal semilinear equation $\partial_t^αu+(-Δ)^βu=|u|^{p-1}u$, $p\ge1$, where $\partial_t^α$ stands for the Caputo derivative of order $α\in (0,1)$ and $(-Δ)^β$, $β\in(0,1]$, is the usual $β$ power of the Laplacian. We prescribe an initial datum in $L^q(\mathbb{R}^N)$.
We give conditions ensuring the existence and uniqueness of a solution living in $L^q(\mathbb{R}^N)$ up to a maximal existence time $T$ that may be finite or infinite. If~$T$ is finite, the $L^q$ norm of the solution becomes unbounded as time approaches $T$, and $u$ is said to blow up in $L^q$. Otherwise, the solution is global in time.
For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. It turns out that every nonnegative nontrivial solution in $L^q$ blows up in finite time if $1<p<p_f:=1+\frac{2β}N$ whereas if $p\ge p_f$ there are both solutions that blow up and global ones. The critical exponent $p_f$, which does not depend on $α$, coincides with the Fujita exponent for the case $α=1$, in which the time derivative is the standard (local) one. In contrast to the case $α=1$, when $α\in(0,1)$ the critical exponent $p=p_f$ falls within the situation in which global existence may occur. Our weakest condition for global existence and our condition for blow-up are both related to the size of the mean value of the initial datum in large balls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A semilinear problem associated to the space-time fractional heat equation in $\mathbb{R}^N$ Cortázar, Carmen Quirós, Fernando Wolanski, Noemí Analysis of PDEs 35B44, 35B33, 35K58, 35R11, 35A01, 35A02, 35B60 We study the fully nonlocal semilinear equation $\partial_t^αu+(-Δ)^βu=|u|^{p-1}u$, $p\ge1$, where $\partial_t^α$ stands for the Caputo derivative of order $α\in (0,1)$ and $(-Δ)^β$, $β\in(0,1]$, is the usual $β$ power of the Laplacian. We prescribe an initial datum in $L^q(\mathbb{R}^N)$. We give conditions ensuring the existence and uniqueness of a solution living in $L^q(\mathbb{R}^N)$ up to a maximal existence time $T$ that may be finite or infinite. If~$T$ is finite, the $L^q$ norm of the solution becomes unbounded as time approaches $T$, and $u$ is said to blow up in $L^q$. Otherwise, the solution is global in time. For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. It turns out that every nonnegative nontrivial solution in $L^q$ blows up in finite time if $1<p<p_f:=1+\frac{2β}N$ whereas if $p\ge p_f$ there are both solutions that blow up and global ones. The critical exponent $p_f$, which does not depend on $α$, coincides with the Fujita exponent for the case $α=1$, in which the time derivative is the standard (local) one. In contrast to the case $α=1$, when $α\in(0,1)$ the critical exponent $p=p_f$ falls within the situation in which global existence may occur. Our weakest condition for global existence and our condition for blow-up are both related to the size of the mean value of the initial datum in large balls. |
| title | A semilinear problem associated to the space-time fractional heat equation in $\mathbb{R}^N$ |
| topic | Analysis of PDEs 35B44, 35B33, 35K58, 35R11, 35A01, 35A02, 35B60 |
| url | https://arxiv.org/abs/2405.18612 |