Let V be a locally finite, connected and weighted graph. In this paper, we establish the existence and non-existence of positive solutions of discrete parabolic inequality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_t-\Delta u\ge u^p \text{ in } V\times I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>≥</mo> <msup> <mi>u</mi> <mi>p</mi> </msup> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi>V</mi> <mo>×</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> and system of discrete parabolic inequalities <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\left\{ \begin{array}{ll} u_t-\Delta u\ge v^p \text{ in } V\times I,\\ v_t-\Delta v\ge u^q \text{ in } V\times I, \end{array}\right. }$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>≥</mo> <msup> <mi>v</mi> <mi>p</mi> </msup> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi>V</mi> <mo>×</mo> <mi>I</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>≥</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi>V</mi> <mo>×</mo> <mi>I</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p,q\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and I is the whole line <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> or a half-line. In particular, we are able to prove non-existence results in the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where the test function method can not be applied.

Liouville-Type Results for a System of Parabolic Inequalities on Locally Finite Graphs

期刊论文
发表日期：

2026年1月21日

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Anh Tuan Duong,
Setsuro Fujiié

摘要

Let V be a locally finite, connected and weighted graph. In this paper, we establish the existence and non-existence of positive solutions of discrete parabolic inequality $u_t-\Delta u\ge u^p \text{ in } V\times I$ $u_{t} - Δ u \geq u^{p} in V \times I$ and system of discrete parabolic inequalities ${\left\{ \begin{array}{ll} u_t-\Delta u\ge v^p \text{ in } V\times I,\\ v_t-\Delta v\ge u^q \text{ in } V\times I, \end{array}\right. }$ $\{\begin{matrix} u_{t} - Δ u \geq v^{p} in V \times I, \\ v_{t} - Δ v \geq u^{q} in V \times I, \end{matrix})$ where $p,q\in \mathbb {R}$ $p, q \in R$ and I is the whole line $\mathbb {R}$ $R$ or a half-line. In particular, we are able to prove non-existence results in the case $p<1$ $p < 1$ or $q<1$ $q < 1$ , where the test function method can not be applied.