*Mathematical Methods in the Applied Sciences*Blowup of sign-changing radial solutions for a semilinear parabolic
equation

This paper is concerned with sign-changing radial solutions of the
semilinear parabolic equation $$\left\{
\begin{gathered}
u_t-u_{rr}-\frac{N-1}{r}u_r=a(r)u+|u|^{p
- 1}u, \ \ \
\ r\in(0,1), t>0,
\hfill \\ u_r(0,t)=0,
\ \ \ \
\ \ \ \
\ \ \ \
u(1,t)=0, \ \ \
\ \ \ \
\ \ \ \
\ t>0 \hfill
\\ \end{gathered}
\right.\leqno{\rm({P})}
$$ with initial data $u(r,0)=u_0(r)$,
$r\in[0,1]$, where $u_0(r)$,
$a(r)\in C[0,1]$, $u_0(r)$ is not identically
equal to $0$ in $[0,1]$, $p>1$,
$N>1$. Under suitable assumptions on
$\lambda_k$, we prove that solutions blowup in finite
time if $z(u_0)\leq k$, while there exist stationary
solutions with $k$ or more zeros, where $\lambda_k$
is the k-th eigenvalue of linearized equation, and
$z(\cdot)$ is the number of times of sign changes.

13 Jul 2020

17 Jul 2020

17 Jul 2020

03 Aug 2020