I am thinking about a geometric problem which boils down to the following parabolic equation:
Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$
$$\begin{cases}\displaystyle \frac{\partial u}{\partial t}=(\ln u)''+\frac{1}{r}(\ln u)'-4u(k\sqrt{u(1)}-1)\\u'(0)=0\\u'(1)=2u(1)(k\sqrt{u(1)}-1)\end{cases}$$
Here $'$ means differentiating respect to $r$. $u(1)$ is $u(1,t)$ for short. $k$ is some nice **positive** constant fixed. You can assume it is very small.

Obviously this is a nonlocal quasilinear equation with nonlinear neumann boundary condition. We always assume that the initial data $u(\cdot,0)>0$. And it has a particular solution which is $u\equiv \frac{1}{k^2}$. A nice property of this equation is $\int_0^1 ru(r,t)dr$ is unchanged along the flow.

I am very interested in the long time existence of this equation. Suppose we have very nice initial data which is positive

(1)Will $u$ exists forever?

(2)If this is true, will $u$ converge to some solution?

(3) What about the stability of this particular solution?

(4)All the above question is linked to examine the behavior of $u(1)$. If one can control $u(1)$, then it is done. Can anyone construct one solution $u$ such that $u(1,t)\to \infty$ when $t\to T$ where $T$ can be finite or infinite? Or $u(1,t)\to 0$ when $t\to T$? Conunterexample is welcomed, because I know every few about this equation.

Thank you for any observation in advance.