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arXiv · 2503.02446

The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities

Reiri Miyamoto·Motohiro Sobajima
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Abstract

This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\ u(x,0)=u_0(x)\ge0,&x\in\mathbb{R}, \end{cases} \end{equation} where $p>1$, $m\ge0$, $u_0\in BC(\mathbb{R})$ and the potential $V=V(x)\in BC(\mathbb{R})$ satisfies a certain property. More precisely, we determine the critical Fujita exponent for (P), that is, the threshold for the global existence/nonexistence of (P).

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Reiri Miyamoto, Motohiro Sobajima. 2025-03-04. The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities. https://arxiv.org/abs/2503.02446

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