<p>We study the large-time asymptotics of global solutions to the semilinear heat equation in <InlineEquation ID="IEq1"> <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="IEq2"> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) with critical Sobolev exponent <Equation ID="Equ49"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+|u|^{\frac{4}{n-2}} u ~& \hbox { in }~ {{{\mathbb {R}}}}^n \times (0,\infty ),\\ u(\cdot ,0)=u_0~& \hbox { in }~ {{{\mathbb {R}}}}^n. \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <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> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mfrac> <mn>4</mn> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </msup> <mi>u</mi> <mspace width="3.33333pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <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> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mspace width="3.33333pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, we construct global and positive solutions for a class of initial value <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_0(x)\sim |x|^{-\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|x|\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma >2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> such that the asymptotics of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Vert u\Vert _{L^\infty ({\mathbb {R}}^6)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>u</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> depends on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> in a precise manner, motivating by a program proposed by Fila and King [<CitationRef CitationID="CR11">11</CitationRef>]. Some remarks on the lower dimensional cases are given.</p>

Some global solutions to the energy-critical semilinear heat equation

期刊论文
发表日期：

2025年6月23日

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Juncheng Wei,
Yifu Zhou

摘要

We study the large-time asymptotics of global solutions to the semilinear heat equation in ${\mathbb {R}}^n$ $R^{n}$ ( $n\ge 3$ $n \geq 3$ ) with critical Sobolev exponent $\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+|u|^{\frac{4}{n-2}} u ~& \hbox { in }~ {{{\mathbb {R}}}}^n \times (0,\infty ),\\ u(\cdot ,0)=u_0~& \hbox { in }~ {{{\mathbb {R}}}}^n. \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u + {| u |}^{\frac{4}{n - 2}} u & in R^{n} \times (0, \infty), \\ u (\cdot, 0) = u_{0} & in R^{n} . \end{matrix}) \end{matrix}$ For $n=6$ $n = 6$ , we construct global and positive solutions for a class of initial value $u_0(x)\sim |x|^{-\gamma }$ $u_{0} (x) \sim {| x |}^{- γ}$ as $|x|\rightarrow \infty $ $| x | \to \infty$ with $\gamma >2$ $γ > 2$ such that the asymptotics of $\Vert u\Vert _{L^\infty ({\mathbb {R}}^6)}$ ${‖ u ‖}_{L^{\infty} (R^{6})}$ depends on $\gamma $ $γ$ in a precise manner, motivating by a program proposed by Fila and King [11]. Some remarks on the lower dimensional cases are given.