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

TitleA Liouville comparison principle for solutions to semilinear parabolic second-order partial differential inequalities in the whole space
Publication TypeJournal Article
Year of Publication2014
AuthorsKurta, VV
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2014.03.036
Issue3
SectionMathematics
Pagination36-42
Date Published3/2014
LanguageEnglish
Abstract
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ to semilinear parabolic second-order partial differential inequalities of the form

$u_{t} -{\mathcal L}u- |u|^{q-1}u\geq v_{t} -{\mathcal L}v- |v|^{q-1}v\tag{$*$}$

in the whole space ${\mathbb R} \times \mathbb R^{n}$. Here, $n\geq1$, $q>1$, 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 $a_{ij}(t,x)$, $i$, $j=1,\ldots ,n$, are functions that are defined, measurable, and locally bounded
in ${\mathbb R} \times \mathbb R^{n}$ 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 $(t,x)\in {\mathbb R} \times \mathbb R^{n}$ 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 ${\mathbb R} \times\mathbb R^{n}$, 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 $u$ $(*)$ in the whole space ${\mathbb R} \times \mathbb R^{n}$ in the case where $v\equiv 0$. All the results obtained are new and sharp.

KeywordsLiouville comparison principle, second-order partial inequalities, space
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