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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.11134 |
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| _version_ | 1866917642381557760 |
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| author | Wei, Juncheng Ye, Zikai Zeng, Xiaoyu Zhang, Qidi |
| author_facet | Wei, Juncheng Ye, Zikai Zeng, Xiaoyu Zhang, Qidi |
| contents | We consider the following five-dimensional heat equation with critical boundary condition \begin{equation*}
\partial_t u=Δu
\mbox{ \ in \ } \mathbb{R}_+^5\times (0,T) ,
\quad
-\partial_{x_5}u =|u|^\frac{2}{3}u \mbox{ \ on \ } \pp \mathbb{R}^5_+ \times (0,T) . \end{equation*} Given $\mathfrak{o}$ distinct boundary points $q^{[i]} \in \partial \mathbb{R}_+^5$, and $\mathfrak{o}$ integers $l_i\in \mathbb{N}$ (possibly duplicated), $i=1,2,\dots, \mathfrak{o}$, for $T>0$ sufficiently small, we construct a finite-time blow-up solution $u$ with a type II blow-up rate $(T-t)^{-3l_i -3}$ for $x$ near $q^{[i]}$. This seems to be the first result of the co-existence of type II blowups with different blow-up rates. To accommodate highly unstable blowups with different blowup rates, we first develop a unified linear theory for the inner problem with more time decay in the blow-up scheme through restriction on the spatial growth of the right-hand side, and then use vanishing adjustment functions for deriving multiple rates at distinct points. This paper is inspired by [25, 52, 60]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions Wei, Juncheng Ye, Zikai Zeng, Xiaoyu Zhang, Qidi Analysis of PDEs We consider the following five-dimensional heat equation with critical boundary condition \begin{equation*} \partial_t u=Δu \mbox{ \ in \ } \mathbb{R}_+^5\times (0,T) , \quad -\partial_{x_5}u =|u|^\frac{2}{3}u \mbox{ \ on \ } \pp \mathbb{R}^5_+ \times (0,T) . \end{equation*} Given $\mathfrak{o}$ distinct boundary points $q^{[i]} \in \partial \mathbb{R}_+^5$, and $\mathfrak{o}$ integers $l_i\in \mathbb{N}$ (possibly duplicated), $i=1,2,\dots, \mathfrak{o}$, for $T>0$ sufficiently small, we construct a finite-time blow-up solution $u$ with a type II blow-up rate $(T-t)^{-3l_i -3}$ for $x$ near $q^{[i]}$. This seems to be the first result of the co-existence of type II blowups with different blow-up rates. To accommodate highly unstable blowups with different blowup rates, we first develop a unified linear theory for the inner problem with more time decay in the blow-up scheme through restriction on the spatial growth of the right-hand side, and then use vanishing adjustment functions for deriving multiple rates at distinct points. This paper is inspired by [25, 52, 60]. |
| title | Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.11134 |