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

We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t-\mathcal{L}u-\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u\ge v_t-\mathcal{L}v-\mathrm{\mid }v{\mathrm{\mid }}^{q-1}v(\ast )$$ in the whole space $\mathbb{E}=\mathbb{R}\times {\mathbb{R}}^n$. Here, $n\geq 1$,$q>0$and$$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ and \sum_{i,j=1}^n a_{ij}(t,x)\xi_i\xi_j\geq 0 $$foralmostall$(t,x)\in \mathbb E$andall$\xi \in \mathbb R^n$. We show that the critical exponents in the Liouville comparison principle obtained, which are responsible for the non-existence of non-trivial (i.e., such that $u\equiv ̸v$) weak solutions to () in the whole space $\mathbb{E}$, depend on the behavior of the coefficients of the operator $\mathcal{L}$ at infinity and coincide with those obtained for solutions of () in the half-space ${\mathbb{R}}_{+}\times {\mathbb{R}}^n$. As direct corollaries we obtain new Liouville-type theorems for non-negative weak solutions $u$ of the inequality (*) in the whole space $\mathbb{E}$ in the case when $v\equiv 0$. All the results obtained are new and sharp.