## Rocky Mountain Journal of Mathematics

### Thermal avalanche after non-simultaneous blow-up in heat equations coupled via nonlinear boundary

#### Abstract

In this paper, we study a parabolic problem defined in the half line coupled via exponential-type boundary flux. Firstly, we prove the optimal classification of the two components of blow-up solutions when time reaches the blow-up time from left. Blow-up takes place only at the origin, and simultaneous blow-up rates are determined as well. Secondly, we study the weak extension after the blow-up time. Complete blow-up always occurs whether simultaneous blow-up arises or not. Moreover, an instantaneous propagation of the blow-up singularity to the whole spatial domain occurs at the blow-up time, which is the so-called thermal avalanche phenomenon. Finally, we use the evolution of the $k$-level set of solutions in the approximations to characterize the propagation of the singularity.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 3 (2019), 817-847.

Dates
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.rmjm/1563847235

Digital Object Identifier
doi:10.1216/RMJ-2019-49-3-817

Mathematical Reviews number (MathSciNet)
MR3983302

Zentralblatt MATH identifier
07088338

#### Citation

Liu, Bingchen; Li, Fengjie; Dong, Mengzhen. Thermal avalanche after non-simultaneous blow-up in heat equations coupled via nonlinear boundary. Rocky Mountain J. Math. 49 (2019), no. 3, 817--847. doi:10.1216/RMJ-2019-49-3-817. https://projecteuclid.org/euclid.rmjm/1563847235

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