We consider the evolutionary p-Laplace equation in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^n\times (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, we construct a solution u with a moving gradient singularity in the sense that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|\nabla u(x,t)|\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> for each t as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x\rightarrow \xi (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\xi :[0,\infty )\rightarrow \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>:</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a given curve.

Moving gradient singularity for the evolutionary p-Laplace equation

期刊论文
发表日期：

2025年4月26日

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Erik Lindgren,
Jin Takahashi

摘要

We consider the evolutionary p-Laplace equation in $\mathbb {R}^n\times (0,\infty )$ $R^{n} \times (0, \infty)$ . For $p>n$ $p > n$ , we construct a solution u with a moving gradient singularity in the sense that $|\nabla u(x,t)|\rightarrow \infty $ $| \nabla u (x, t) | \to \infty$ for each t as $x\rightarrow \xi (t)$ $x \to ξ (t)$ , where $\xi :[0,\infty )\rightarrow \mathbb {R}^n$ $ξ : [0, \infty) \to R^{n}$ is a given curve.