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

TL;DR

This paper establishes a new Liouville comparison principle for weak solutions of semilinear parabolic inequalities in the entire space, revealing how the critical exponents depend on the operator's coefficients at infinity and extending known results.

Contribution

It introduces a novel Liouville comparison principle for weak solutions of semilinear parabolic inequalities, accounting for variable coefficients and their behavior at infinity, with sharp, new results.

Findings

Critical exponents depend on coefficients at infinity.

Non-existence of non-trivial solutions under certain conditions.

New Liouville-type theorems for non-negative solutions.

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.