Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Liouville comparison principle for solutions of semilinear parabolic
  second-order partial differential inequalities   We obtain a new Liouville comparison principle for entire weak solutions
$(u,v)$ of semilinear parabolic second-order partial differential inequalities
of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v-
|v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times
\mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal
L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x)
\frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are
functions defined, measurable and locally bounded in $\mathbb S$, and such that
$a_{ij}(t,x)=a_{ji}(t,x)$ and $$ \sum_{i,j=1}^n a_{ij}(t,x)\xi_i\xi_j\geq 0 $$
for almost all $(t,x)\in \mathbb S$ and all $\xi \in \mathbb R^n$. The critical
exponents in the Liouville comparison principle obtained, which responsible for
the non-existence of non-trivial (i.e., such that $u\not \equiv v$) entire weak
solutions to (*) in $\mathbb S$, depend on the behaviour of the coefficients of
the operator $\mathcal L$ at infinity. As direct corollaries we obtain a new
Fujita comparison principle for entire weak solutions $(u,v)$ of the Cauchy
problem for the inequality (*), as well as new Liouville-type and Fujita-type
theorems for non-negative entire weak solutions $u$ of the inequality (*) in
the case when $v\equiv 0$. All the results obtained are new and sharp.

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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph