Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We are concerned with a Cauchy problem of the semilinear heat equation <Equation ID="Equ46"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu-\Delta u=f(u), & x\in \mathbb {R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in \mathbb {R}^N, \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <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>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, f is nonnegative, increasing and convex, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\log f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is convex for large <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and some additional assumptions are assumed. We establish a positive radial singular stationary solution <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u^(x)\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$|x|\rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, we prove the following: The problem has a nonnegative global-in-time solution if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$0\le u_0\le u^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>≤</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$u_0\not \equiv u^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>≢</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, while the problem has no nonnegative local-in-time solutions u such that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$u\ge u^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>≥</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$u_0\ge u^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>≥</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$u_0\not \equiv u^*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>≢</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>.

Threshold property of a singular stationary solution for semilinear heat equations with exponential growth

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2025年8月18日

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Kotaro Hisa,
Yasuhito Miyamoto

摘要

Let $N\ge 3$ $N \geq 3$ . We are concerned with a Cauchy problem of the semilinear heat equation $\begin{aligned} \left\{ \begin{array}{ll} \partial _tu-\Delta u=f(u), & x\in \mathbb {R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in \mathbb {R}^N, \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} \partial_{t} u - Δ u = f (u), & x \in R^{N}, t > 0, \\ u (x, 0) = u_{0} (x), & x \in R^{N}, \end{matrix}) \end{matrix}$ where $f(0)=0$ $f (0) = 0$ , f is nonnegative, increasing and convex, $\log f(u)$ $log f (u)$ is convex for large $u>0$ $u > 0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ $u^{*}$ such that $u^*(x)\rightarrow \infty $ $u^{*} (x) \to \infty$ as $|x|\rightarrow 0$ $| x | \to 0$ . Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ $0 \leq u_{0} \leq u^{*}$ and $u_0\not \equiv u^*$ $u_{0} ≢ u^{*}$ , while the problem has no nonnegative local-in-time solutions u such that $u\ge u^*$ $u \geq u^{*}$ if $u_0\ge u^*$ $u_{0} \geq u^{*}$ and $u_0\not \equiv u^*$ $u_{0} ≢ u^{*}$ .