Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Entire and ancient solutions of a supercritical semilinear heat equation   We consider the semilinear heat equation $u_t=\Delta u+u^p$ on ${\mathbb
R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical
exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions
defined for all $t\in {\mathbb R}$) and ancient solutions (classical solutions
defined on $(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type
theorem saying that if $p$ is greater than the Lepin exponent $p_L:=1+6/(N-10)$
($p_L=\infty$ if $N\le 10$), then all positive bounded radial entire solutions
are steady states. The theorem is not valid without the assumption of radial
symmetry; in other ranges of supercritical $p$ it is known not to be valid even
in the class of radial solutions. Our other results include classification
theorems for nonstationary entire solutions (when they exist) and ancient
solutions, as well as some applications in the theory of blowup of solutions.

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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: eess