Oscillatory behavior of solutions to the critical Fujita equation in 6D
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915632710156288 |
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| author | Harada, Junichi |
| author_facet | Harada, Junichi |
| contents | Long time dynamics of solutions to the 6D energy critical heat equation
$u_t=Δu+|u|^{p-1}u$ on $\R^6\times(0,\infty)$ is investigated.
It is shown that
there exists a radially symmetric global solution $u(x,t)\in C([0,\infty);\dot H^1(\R^6))$
of the form
\begin{align*}
u(x,t)
=
λ(t)^{-\frac{n-2}{2}}
{\sf Q}(\tfrac{x}{λ(t)})
+
\text{error}
(x,t),
\end{align*}
where the function \( λ(t) \) satisfies:
\begin{itemize}
\item
$\dis\lim_{t\to\infty}\|\text{error}(\cdot,t)\|_{\dot H_x^1(\R^6)}=0$,
\item
$\dis\liminf_{t\to\infty}λ(t)=0$,
\item
$\dis\limsup_{t\to\infty}λ(t)=\infty$.
\end{itemize}
The solutions constructed here
demonstrate that the dynamical behavior in
\( \dot H^1(\mathbb{R}^n) \) can differ significantly
from the behavior in \( H^1(\mathbb{R}^n) \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oscillatory behavior of solutions to the critical Fujita equation in 6D Harada, Junichi Analysis of PDEs 35K58, 35B33, 35B40, 35B44 Long time dynamics of solutions to the 6D energy critical heat equation $u_t=Δu+|u|^{p-1}u$ on $\R^6\times(0,\infty)$ is investigated. It is shown that there exists a radially symmetric global solution $u(x,t)\in C([0,\infty);\dot H^1(\R^6))$ of the form \begin{align*} u(x,t) = λ(t)^{-\frac{n-2}{2}} {\sf Q}(\tfrac{x}{λ(t)}) + \text{error} (x,t), \end{align*} where the function \( λ(t) \) satisfies: \begin{itemize} \item $\dis\lim_{t\to\infty}\|\text{error}(\cdot,t)\|_{\dot H_x^1(\R^6)}=0$, \item $\dis\liminf_{t\to\infty}λ(t)=0$, \item $\dis\limsup_{t\to\infty}λ(t)=\infty$. \end{itemize} The solutions constructed here demonstrate that the dynamical behavior in \( \dot H^1(\mathbb{R}^n) \) can differ significantly from the behavior in \( H^1(\mathbb{R}^n) \). |
| title | Oscillatory behavior of solutions to the critical Fujita equation in 6D |
| topic | Analysis of PDEs 35K58, 35B33, 35B40, 35B44 |
| url | https://arxiv.org/abs/2511.17891 |