<p>The paper is concerned with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\partial _tu-\Delta u+u^p=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>u</mi> <mi>p</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in exterior domains <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>). The problem for the one-dimensional case is formulated with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega =(0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which is one of the representative of the connected components in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. One can see that the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroup for the corresponding linear problem possesses an invariant measure <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\phi (x)\,dx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\phi (x)\,dx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> vanishes as <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$1<p\le \min \{2,1+\frac{2}{N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>2</mn> <mi>N</mi> </mfrac> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In the other case <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$p>\min \{2,1+\frac{2}{N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>2</mn> <mi>N</mi> </mfrac> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains

期刊论文
发表日期：

2026年5月24日

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Ahmad Fino,
Motohiro Sobajima

摘要

The paper is concerned with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial _tu-\Delta u+u^p=0$ $\partial_{t} u - Δ u + u^{p} = 0$ in exterior domains $\Omega $ $Ω$ in $\mathbb {R}^N$ $R^{N}$ ( $N\ge 2$ $N \geq 2$ ). The problem for the one-dimensional case is formulated with $\Omega =(0,\infty )$ $Ω = (0, \infty)$ which is one of the representative of the connected components in $\mathbb {R}$ $R$ . One can see that the $C_0$ $C_{0}$ -semigroup for the corresponding linear problem possesses an invariant measure $\phi (x)\,dx$ $ϕ (x) d x$ , where $\phi $ $ϕ$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $\phi (x)\,dx$ $ϕ (x) d x$ vanishes as $t\rightarrow \infty $ $t \to \infty$ if and only if $1<p\le \min \{2,1+\frac{2}{N}\}$ $1 < p \leq min {2, 1 + \frac{2}{N}}$ . In the other case $p>\min \{2,1+\frac{2}{N}\}$ $p > min {2, 1 + \frac{2}{N}}$ , we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when $N\ge 3$ $N \geq 3$ .