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We obtain a new Liouville comparison principle for weak solutions $(u,v)$ ofsemilinear parabolic second-order partial differential inequalities of the form$$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ inthe whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq1$, $q>0$ and $$ {\mathcalL}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x)\frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$,are functions that are defined, measurable and locally bounded in $\mathbb E$,and such that $a_{ij}(t,x)=a_{ji}(t,x)$ and $$ \sum_{i,j=1}^na_{ij}(t,x)\xi_i\xi_j\geq 0 $$ for almost all $(t,x)\in \mathbb E$ and all $\xi\in \mathbb R^n$. We show that the critical exponents in the Liouvillecomparison principle obtained, which are responsible for the non-existence ofnon-trivial (i.e., such that $u\not \equiv v$) weak solutions to (*) in thewhole space $\mathbb E$, depend on the behavior of the coefficients of theoperator $\mathcal L$ at infinity and coincide with those obtained forsolutions of (*) in the half-space ${\mathbb R}_+\times {\mathbb R}^n$. Asdirect corollaries we obtain new Liouville-type theorems for non-negative weaksolutions $u$ of the inequality (*) in the whole space $\mathbb E$ in the casewhen $v\equiv 0$. All the results obtained are new and sharp.

A Liouville comparison principle for weak solutions of semilinear parabolic second-order partial differential inequalities in the whole space