Remarks to the blow-up rate of a degenerate parabolic equation (Q2708583)

This is sharpening the following Wiegner's result concerning blow-up behaviour of solution of the equation NEWLINE\[NEWLINE u_t=u^p (u_{xx}+u)\text{ on } (-a, a)\times (0,T). \tag{1} NEWLINE\]NEWLINE Let \(a>\frac\pi 2, \varphi:[-a,a]\to \mathbb{R} \) a \(C_3\) -- function with: \(\varphi(x)>0,\) \(\varphi(x)=\varphi(-x)\) and \(x\varphi'(x)\leq 0\) on \((-a,a), \varphi(\pm a)=0\), \(\varphi''(x) + \varphi(x)\geq 0\), \(x \cdot (\varphi'''(x) + \varphi'(x))\leq 0 \) for \(x\in(-a,a)\), \(\pm\varphi'(\pm a)<0\). Then there is a finite time \(T>0,\) such that a unique solution \( u \) of (1) exists with \(u(\pm a, t)=0, u(x,0)=\varphi(x)\). In particular it has the following properties: \(u(x)>0, u_t(x,t)\geq 0, u(x,t)=u(-x,t), u(0,t) = \underset{|x|\leq a}\max u(x,t)\); \(p u^p(0,t) (T-t)\geq 1 \) for \( t<T\); the profile \((u(0,t))^{-1} u(x,t)\) is monotonically decreasing for \(t\to T\) with continuous limit function NEWLINE\[NEWLINEL(x)=\begin{cases} \cos x, |x|\leq\frac\pi 2 \\ 0, |x|>\frac\pi 2;\end{cases}NEWLINE\]NEWLINE if \(p\geq 2\), then \(\lim_{t\to T}(pu(0,t)^p(T-t))^{-1}=0\), \(\overline S=[-\frac\pi 2, \frac\pi 2]\) where \(S=\{x:u(x,t)\to\infty\) for \(t\to T\}\) is the blow-up set.NEWLINENEWLINENEWLINETheorem. Let \(p=2\). Then, for arbitrary \(\varepsilon>0\) there is some \(C_\varepsilon>0\), such that NEWLINE\[NEWLINE\begin{aligned} \left(2(T-t)\right)^{-\frac{1}{2}} & \leq u(0,t) \leq C_{\varepsilon} (T-t)^{-\frac{1}{2}-\varepsilon},\\ \pi \left(2(T-t)\right)^{-\frac{1}{2}} & \geq u(\frac\pi 2,t) \geq C_\varepsilon^{-1}(T-t)^{-\frac 12+\varepsilon}. \end{aligned}NEWLINE\]NEWLINE{}.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].