arXiv subjects

Vasilii V. Kurta

Publications and source records attributed to Vasilii V. Kurta.

6 recordsLinked to original sources

arXiv2021

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

We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are defined and measurable in Rn and which belong locally to a Sobolev-type function space also associated with the operator L.

math.AP↗

arXiv2013

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 $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ 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 $$ \sum_{i,j=1}^n a_{ij}(t,x)ξ_iξ_j\geq 0 $$ for almost all $(t,x)\in \mathbb E$ and all $ξ\in \mathbb R^n$. 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.

math.AP↗

arXiv2012

A Liouville comparison principle for sub- and super-solutions of the equation $w_t-Δ_p (w) = |w|^{q-1}w$

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation $(\ast)$ $w_t-Δ_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ Δ_p (w):={div}_x(|\nabla_x w|^{p-2}\nabla_x w)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entire weak super-solution and an entire weak sub-solution of ($\ast$) in $\Bbb S$ which belong, only locally in $\Bbb S$, to the corresponding Sobolev space and are such that $u\geq v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.

math.AP↗

arXiv2012

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

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 $$ \sum_{i,j=1}^n a_{ij}(t,x)ξ_iξ_j\geq 0 $$ for almost all $(t,x)\in \mathbb S$ and all $ξ\in \mathbb R^n$. 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.

math.AP↗

arXiv2011

A Liouville comparison principle for solutions of quasilinear differential inequalities

This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form $A(u) + |u|^{q-1}u \leq A(v) + |v|^{q-1}v$ on ${\Bbb R}^n$, where $n\geq 1$, $q$ is positive, and the operator $A(w)$ belongs to a class of the so-called $α$-monotone operators. Typical examples of such operators are the $p$-Laplacian and its well-known modifications. The results improve and supplement those in [1].

math.AP↗

arXiv2011

A Liouville comparison principle for entire sub- and super-solutions of the equation $u_t-Δ_p (u) = |u|^{q-1}u$

We establish a Liouville comparison principle for entire sub- and super-solutions of the equation $(\ast)$ $w_t-Δ_p (w) = |w|^{q-1}w$ in the half-space ${\mathbb S}= {\mathbb R}^1_+\times {\mathbb R}^n$, where $n\geq 1$, $q>0$ and $ Δ_p (w):={div}_x(|\nabla_x w|^{p-2}\nabla_x w)$, $1<p\leq 2$. In our study we impose neither restrictions on the behaviour of entire sub- and super-solutions on the hyper-plane $t=0$, nor any growth conditions on the behavior of them or any of their partial derivatives at infinity. We prove that if $1<q\leq p-1+\frac pn$, and $u$ and $v$ are, respectively, an entire weak super- and an entire weak sub-solution of ($\ast$) in $\Bbb S$ which belong, only locally in $\Bbb S$, to the corresponding Sobolev space and are such that $u\leq v$, then $u\equiv v$. The result is sharp. As direct corollaries we obtain both new and known Fujita-type and Liouville-type results.

math.AP↗