Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Prevention of blowup via Neumann heat kernel   Consider the heat equation $u_t-\Delta u=0$ on a bounded $C^2$ domain
$\Omega$ in $\mathbb{R}^{n}(n\geq 2)$ with any positive initial data. If a
superlinear radiation law $\frac{\partial u}{\partial n}=u^{q}$ with $q>1$ is
imposed on a partial boundary $\Gamma_1\subseteq\partial\Omega$ which has a
positive surface area, then it has been known that the solution $u$ blows up in
finite time. However, if the partial boundary, on which the superlinear
radiation law is prescribed, is shrinking and is denoted as $\Gamma_{1,t}$ at
time $t$, then the solution may exist globally as long as the surface area
$|\Gamma_{1,t}|$ of $\Gamma_{1,t}$ decays fast enough. This paper asks the
question that how fast should $|\Gamma_{1,t}|$ decay in order to have a bounded
global solution? This question is of significant importance in realistic
situations, such as the temperature control within a certain safe range. By
taking advantage of the Neumann heat kernel, we conclude that a polynomial
decay $|\Gamma_{1,t}|\sim |\Gamma_1|(1+Ct)^{-\beta}$ with any $\beta>n-1$
suffices to ensure a bounded global solution.

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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph