Blow-up of positive solutions for the semilinear heat equation with a potential

Abstract

We study nonnegative solutions of the equation $u_t-\mathrm{\Delta }u+V\left(x\right)u=|u{|}^{p-1}u$ in ${\mathbb{R}}^n$, t>0, under the assumption that $V\left(x\right)\in C^1\left({\mathbb{R}}^n\right)$ satisfies $|V\left(x\right)|\le c$. We establish a new blow-up criterion that depends only on $V\left(x\right)$ and p. In addition, as the supreme of $V\left(x\right)$ decreases, we find an interesting phenomenon that the Fujita exponent goes to ∞, in the sense that every nonnegative solution blows up in finite time whenever p>1. Furthermore, we obtain the blow-up rate estimate in the subcritical case. In the end, under the assumption of $V\left(x\right)\ge 0$, we give the refined blow-up estimate of the blow-up solutions near the blow-up time.

KEYWORDS:

• Heat equation
• Fujita exponent
• blow-up rate
• blow-up set

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35K05
• 35B40
• 35B44

Acknowledgements

The author would like to thank Professor Li Ma for his helpful suggestions and discussions. We thank the referee who gives valuable comments and useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China No. 11771124 and a research grant from USTB, China.