Electron. J. Diff. Equ., Vol. 2015 (2015), No. 311, pp. 1-13.

Quenching behavior of semilinear heat equations with singular boundary conditions

Burhan Selcuk, Nuri Ozalp

Abstract:
In this article, we study the quenching behavior of solution to the semilinear heat equation
$$ v_t=v_{xx}+f(v), $$
with $f(v)=-v^{-r}$ or $(1-v)^{-r}$ and
$$ v_x(0,t)=v^{-p}(0,t), \quad v_x(a,t) =(1-v(a,t))^{-q}. $$
For this, we utilize the quenching problem $u_t=u_{xx}$ with $u_x(0,t)=u^{-p}(0,t)$, $u_x(a,t)=(1-u(a,t))^{-q}$. In the second problem, if $u_0$ is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is $x=0$ ( $x=a$) and $u_t$ blows up at quenching time. Further, we obtain a local solution by using positive steady state. In the first problem, we first obtain a local solution by using monotone iterations. Finally, for $f(v)=-v^{-r}$ ( $(1-v)^{-r}$), if $v_0$ is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is $x=0$ ( $x=a$) and $v_t$ blows up at quenching time.

Submitted October 16, 2015. Published December 21, 2015.
Math Subject Classifications: 35K05, 35K15, 35B50.
Key Words: Heat equation; singular boundary condition; quenching; maximum principle; monotone iteration.

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Burhan Selcuk
Department of Computer Engineering
Karabuk University
Bali klarkayasi Mevkii 78050, Turkey
email: bselcuk@karabuk.edu.tr, burhanselcuk44@gmail.com
Nuri Ozalp
Department of Mathematics, Ankara University
Besevler 06100, Turkey
email: nozalp@science.ankara.edu.tr

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