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

Abstract

We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ 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 $$ \sum_{i,j=1}^n a_{ij}(t,x)\xi_i\xi_j\geq 0 $$ for almost all $(t,x)\in \mathbb E$ and all $\xi \in \mathbb R^n$. 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 $u\not \equiv v$) 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.