## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 17, Number 5 (2010), 891-897.

### A note on blow-up of a nonlinear integral equation

#### Abstract

Let us deal with the positive solutions of \begin{equation*} \frac{\partial u(t)}{\partial t}=k(t)\Delta _{\alpha }u(t)+h(t)u^{1+\beta }(t),\text{ \ }u(0,x)=\varphi (x)\geq 0,\text{ }x\in \mathbb{R}^{d}, \end{equation*} where $\Delta _{\alpha }$ is the fractional Laplacian, $0<\alpha \leq 2$, and $\beta >0$ is a constant. We prove that under certain regularity condition on $\varphi $, $h$ and $k$ any non-trivial positive solution blows up in finite time. In this way we answer, in particular, the question raised in [4] for the critical case.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 5 (2010), 891-897.

**Dates**

First available in Project Euclid: 14 December 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1292334063

**Digital Object Identifier**

doi:10.36045/bbms/1292334063

**Mathematical Reviews number (MathSciNet)**

MR2777778

**Zentralblatt MATH identifier**

1223.35090

**Subjects**

Primary: 35K55: Nonlinear parabolic equations 35K20: Initial-boundary value problems for second-order parabolic equations

Secondary: 35K57: Reaction-diffusion equations 35B35: Stability

**Keywords**

Blow-up of semilinear equations critical dimension blow-up in finite time

#### Citation

Pérez, A.; Villa, J. A note on blow-up of a nonlinear integral equation. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 891--897. doi:10.36045/bbms/1292334063. https://projecteuclid.org/euclid.bbms/1292334063