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\begin{document}

\title{Blowup of sign-changing radial solutions for a semilinear parabolic
equation}



\author[1]{Linfeng Luo}%
\affil[1]{Xi'an Jiaotong University}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\begin{abstract}
This paper is concerned with sign-changing radial solutions of the
semilinear parabolic equation \$\$\textbackslash{}left\textbackslash{}\{
\textbackslash{}begin\{gathered\}
u\_t-u\_\{rr\}-\textbackslash{}frac\{N-1\}\{r\}u\_r=a(r)u+\textbar{}u\textbar{}\^{}\{p
- 1\}u, \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} r\textbackslash{}in(0,1), t\textgreater{}0,
\textbackslash{}hfill \textbackslash{}\textbackslash{} u\_r(0,t)=0,
\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{}
u(1,t)=0, \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} \textbackslash{} \textbackslash{} \textbackslash{}
\textbackslash{} t\textgreater{}0 \textbackslash{}hfill
\textbackslash{}\textbackslash{} \textbackslash{}end\{gathered\}
\textbackslash{}right.\textbackslash{}leqno\{\textbackslash{}rm(\{P\})\}
\$\$ with initial data \$u(r,0)=u\_0(r)\$,
\$r\textbackslash{}in{[}0,1{]}\$, where \$u\_0(r)\$,
\$a(r)\textbackslash{}in C{[}0,1{]}\$, \$u\_0(r)\$ is not identically
equal to \$0\$ in \${[}0,1{]}\$, \$p\textgreater{}1\$,
\$N\textgreater{}1\$. Under suitable assumptions on
\$\textbackslash{}lambda\_k\$, we prove that solutions blowup in finite
time if \$z(u\_0)\textbackslash{}leq k\$, while there exist stationary
solutions with \$k\$ or more zeros, where \$\textbackslash{}lambda\_k\$
is the k-th eigenvalue of linearized equation, and
\$z(\textbackslash{}cdot)\$ is the number of times of sign changes.%
\end{abstract}%



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