We examine the following plate equation with a logarithmic non-linearity term: <Equation ID="Equ26"> <EquationSource Format="TEX">$\begin{aligned} u_{tt}+\Delta ^{2} u+|u_{t}|^{m-2}\,u_{t}=|u|^{p-2}\,u\, \ln |u|^{k}, \quad x \in \Omega , \quad t>0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mrow> <mi>u</mi> <mo>+</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em" /> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em" /> <mi>u</mi> <mspace width="0.166667em" /> <mo>ln</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> </msup> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \Omega \subset {\mathbb {R}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$( n\ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a bounded domain with a smooth boundary, k is a positive constant, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$2< m < p.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>m</mi> <mo><</mo> <mi>p</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> For any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>m> 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>m</mi> <mo>></mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we prove that the blow-up occurs in finite time for arbitrary positive initial energy and suitable initial data.

Blow up phenomenon for a plate equation with logarithmic source term and positive initial energy

期刊论文
发表日期：

2025年5月18日

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Khadijeh Baghaei

摘要

We examine the following plate equation with a logarithmic non-linearity term: $\begin{aligned} u_{tt}+\Delta ^{2} u+|u_{t}|^{m-2}\,u_{t}=|u|^{p-2}\,u\, \ln |u|^{k}, \quad x \in \Omega , \quad t>0, \end{aligned}$ $\begin{matrix} u_{tt} + Δ^{2} u + | u_{t} |^{m - 2} u_{t} = {| u |}^{p - 2} u ln {| u |}^{k}, x \in Ω, t > 0, \end{matrix}$ where $ \Omega \subset {\mathbb {R}}^{n}$ $Ω \subset R^{n}$ $( n\ge 1)$ $(n \geq 1)$ is a bounded domain with a smooth boundary, k is a positive constant, and $2< m < p.$ $2 < m m> 2,$ $p > m > 2,$ we prove that the blow-up occurs in finite time for arbitrary positive initial energy and suitable initial data.