Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910042023788544 |
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| author | Hisa, Kotaro Miyamoto, Yasuhito |
| author_facet | Hisa, Kotaro Miyamoto, Yasuhito |
| contents | We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}(\mathbb{R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^γ_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le γ<N(p_f-1)/2$, where $p_f:=\lim_{u\to\infty}uf'(u)/f(u)$ 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 $f(u)=u^p$ and $e^u$ play a crucial role. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_27098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance Hisa, Kotaro Miyamoto, Yasuhito Analysis of PDEs 35K15, 35A02, 35A21, 35B44 We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}(\mathbb{R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^γ_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le γ<N(p_f-1)/2$, where $p_f:=\lim_{u\to\infty}uf'(u)/f(u)$ 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 $f(u)=u^p$ and $e^u$ play a crucial role. |
| title | Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance |
| topic | Analysis of PDEs 35K15, 35A02, 35A21, 35B44 |
| url | https://arxiv.org/abs/2510.27098 |