Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradient estimates and Liouville type theorems for Poisson equations   In this paper, we will address to the following parabolic equation $$
u_t=\Delta_fu + F(u) $$ on a smooth metric measure space with Bakry-\'{E}mery
curvature bounded from below. Here $F$ is a differentiable function defined in
$\mathbb{R}$. Our motivation is originally inspired by gradient estimates of
Allen-Cahn and Fisher equations (\cite{Bai17, CLPW17}). In this paper, we show
new gradient estimates for these equations. As their applications, we obtain
Liouville type theorems for positive or bounded solutions to the above equation
when either $F=cu(1-u)$ (the Fisher equation) or; $F=-u^3+u$ (the Allen-Cahn
equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).

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The categories of this paper are: q-bio
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The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cs