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

TL;DR

This work analyzes the one-dimensional Cauchy problem $\partial_t u-\partial_x^2 u+V u=\langle x\rangle^{-m}u^p$ with $p>1$ and $m\ge0$, where the potential $V$ is tied to an even function $\psi$ via $V=\psi''/\psi$ and $\psi$ exhibits a power-like growth $\psi(x)\sim\langle x\rangle^{\alpha}$. It develops a framework combining semigroup decay for the Schrödinger-type operator $L=-\partial_x^2+V$, Nash-type inequalities, and a blow-up vs global existence analysis to identify a critical Fujita exponent $p_*(\alpha,m)$ governing global behavior. The results distinguish subcritical, critical, and supercritical regimes and provide explicit criteria for nonexistence in several parameter ranges, as well as constructive global solutions in the supercritical regime via a supersolution approach and decay estimates. The analysis advances understanding of how space-dependent nonlinearities and potentials influence global dynamics and threshold phenomena, with implications for recurrence effects in Brownian motion on weighted manifolds and related Schrödinger-operator criticality theory.

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).

Theorems & Definitions (38)

• Remark 1.1
• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof