On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart
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| Format: | Preprint |
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2023
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| author | Hisa, Kotaro Kojima, Mizuki |
| author_facet | Hisa, Kotaro Kojima, Mizuki |
| contents | We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$.
\[
{\rm (P)}_α\qquad
\partial_t^αu -Δu = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad
u(0) = μ\quad \mbox{in}\quad {\bf R}^N,
\]
where $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$, $p>1$, and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. The case $α=1$ formally gives the Fujita-type equation (P)$_1$ \ $\partial_tu-Δu=u^p$. In particular, we mainly focus on the Fujita critical case where $p=p_F:=1+2/N$. It is well known that the Fujita exponent $p_F$ separates the ranges of $p$ for the global-in-time solvability of (P)$_1$. In particular, (P)$_1$ with $p=p_F$ possesses no global-in-time solutions, and does not locally-in-time solvable in its scale critical space $L^1(\mathbf{R}^N)$. It is also known that the exponent $p_F$ plays the same role for the global-in-time solvability for (P)$_α$. However, the problem (P)$_α$ with $p=p_F$ is globally-in-time solvable, and exhibites local-in-time solvability in its scale critical space $L^1(\mathbf{R}^N)$. The purpose of this paper is to clarify the collapse of the global and local-in-time solvability of (P)$_α$ as $α$ approaches $1-0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16491 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart Hisa, Kotaro Kojima, Mizuki Analysis of PDEs Primary 35R11, Secondary 35K15 We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$. \[ {\rm (P)}_α\qquad \partial_t^αu -Δu = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad u(0) = μ\quad \mbox{in}\quad {\bf R}^N, \] where $\partial_t^α$ denotes the Caputo derivative of order $α\in (0,1)$, $p>1$, and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. The case $α=1$ formally gives the Fujita-type equation (P)$_1$ \ $\partial_tu-Δu=u^p$. In particular, we mainly focus on the Fujita critical case where $p=p_F:=1+2/N$. It is well known that the Fujita exponent $p_F$ separates the ranges of $p$ for the global-in-time solvability of (P)$_1$. In particular, (P)$_1$ with $p=p_F$ possesses no global-in-time solutions, and does not locally-in-time solvable in its scale critical space $L^1(\mathbf{R}^N)$. It is also known that the exponent $p_F$ plays the same role for the global-in-time solvability for (P)$_α$. However, the problem (P)$_α$ with $p=p_F$ is globally-in-time solvable, and exhibites local-in-time solvability in its scale critical space $L^1(\mathbf{R}^N)$. The purpose of this paper is to clarify the collapse of the global and local-in-time solvability of (P)$_α$ as $α$ approaches $1-0$. |
| title | On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart |
| topic | Analysis of PDEs Primary 35R11, Secondary 35K15 |
| url | https://arxiv.org/abs/2307.16491 |