A Liouville comparison principle for solutions of semilinear parabolic second-order partial differential inequalities

Abstract

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.