SOLUTION PROFILES BEYOND QUENCHING FOR SINGULAR SEMILINEAR PARABOLIC PROBLEMS

TitleSOLUTION PROFILES BEYOND QUENCHING FOR SINGULAR SEMILINEAR PARABOLIC PROBLEMS
Publication TypeJournal Article
Year of Publication2018
AuthorsBOONKLURB RATINAN, KAEWRAK KEERATI, TREEYAPRASERT TAWIKAN
JournalDynamic Systems and Applications
Volume27
Issue3
Start Page673
Pagination18
Date Published08/2018
ISSN1056-2176
AMS Subject Classification35K35, 35K57, 35K61
Abstract

Let $T\leq\infty,~a>0,$ $0<r<1,~D=(0,a),~\Omega=D\times(0,T]$ and $\chi(S)$ be the characteristic function of the set $S$. This article studies the steady-state solution after quenching has occured of the semilinear parabolic equation with singularity
\begin{align*}
&  u_{t}-u_{xx}-\frac{r}{x}u_{x}=f(u)\chi(\{u<c\})~\text{in}~\Omega,\\
&  u(x,0)=0~\text{on}~\bar{D},\\
&  u(0,t)=0=u(a,t)~\text{for}~0<t<T.
\end{align*}

It is shown that as $t$ tends to $\infty$, all weak solutions $u\left(x,t\right)$ tend to a unique steady-state solution $U(x)\text{ for }\ 0<{{b}_{s}^{\ast}}\leq x\leq{{B}_{s}^{\ast}}<a$. The numerical methods are developed for computing ${{b}_{s}^{\ast}}$ and ${{B}_{s}^{\ast}}$.

DOI10.12732/dsa.v27i3.15
Refereed DesignationRefereed
Full Text

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