We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider <Equation ID="Equ62"> <EquationSource Format="TEX">$ {\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. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We assume that the growth rate of f is less than the Joseph–Lundgren exponent for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N>10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>></mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> and it satisfies certain assumptions guaranteeing the existence of a positive radial singular stationary solution <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>. We prove that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_0=u^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, then the problem has at least two positive solutions, namely <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and u(t) which satisfies <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u(t)\in L_{loc}^{\infty }(0,t_0;L^{\infty }(\mathbb {R}^N))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mi>∞</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>;</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t_0>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <Equation ID="Equ63"> <EquationSource Format="TEX">$ u(t)\rightarrow u^\quad \text {in}\ L^{\gamma }_{ul}(\mathbb {R}^N)\quad \text {as}\ t\rightarrow 0^+ $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>u</mi> <mo>∗</mo> </msup> <mspace width="1em" /> <mtext>in</mtext> <mspace width="4pt" /> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">ul</mi> </mrow> <mi>γ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mtext>as</mtext> <mspace width="4pt" /> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1\le \gamma <N(p_f-1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>γ</mi> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>f</mi> </msub> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$p_f:=\lim _{u\rightarrow \infty }uf'(u)/f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>f</mi> </msub> <mo>:</mo> <mo>=</mo> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>u</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <mi>u</mi> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a growth rate of f. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of u(t) is based on the monotonicity argument. Transformations of forward self-similar solutions for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$f(u)=u^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$e^u$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mi>u</mi> </msup> </math></EquationSource> </InlineEquation> play a crucial role.

Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

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2026年3月26日

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Kotaro Hisa,
Yasuhito Miyamoto

摘要

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider $ {\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. } $ $\{\begin{matrix} \partial_{t} u - Δ u = f (u), & x \in R^{N}, t > 0, \\ u (x, 0) = u_{0} (x), & x \in R^{N}, \end{matrix})$ where $N>2$ $N > 2$ . We assume that the growth rate of f is less than the Joseph–Lundgren exponent for $N>10$ $N > 10$ and it satisfies certain assumptions guaranteeing the existence of a positive radial singular stationary solution $u^*$ $u^{*}$ . We prove that if $u_0=u^*$ $u_{0} = u^{*}$ , then the problem has at least two positive solutions, namely $u^*$ $u^{*}$ and u(t) which satisfies $u(t)\in L_{loc}^{\infty }(0,t_0;L^{\infty }(\mathbb {R}^N))$ $u (t) \in L_{loc}^{\infty} (0, t_{0}; L^{\infty} (R^{N}))$ for some $t_0>0$ $t_{0} > 0$ and $ u(t)\rightarrow u^*\quad \text {in}\ L^{\gamma }_{ul}(\mathbb {R}^N)\quad \text {as}\ t\rightarrow 0^+ $ $u (t) \to u^{*} in L_{ul}^{γ} (R^{N}) as t \to 0^{+}$ for $1\le \gamma <N(p_f-1)/2$ $1 \leq γ < N (p_{f} - 1) / 2$ , where $p_f:=\lim _{u\rightarrow \infty }uf'(u)/f(u)$ $p_{f} : = {lim}_{u \to \infty} u f^{'} (u) / f (u)$ is a growth rate of f. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of u(t) is based on the monotonicity argument. Transformations of forward self-similar solutions for $f(u)=u^p$ $f (u) = u^{p}$ and $e^u$ $e^{u}$ play a crucial role.