Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equation with critical nonlinearity (Q2748118)

There are described at a formal level some mechanisms of singularity formation for the semilinear parabolic equation NEWLINE\[NEWLINE u_t = \Delta u + |u|^{p-1} u, \;\;u(x,0) = u_0(x), \quad x\in\mathbb{R}^N, \;t>0, NEWLINE\]NEWLINE with critical nonlinearity \(p = p_{\text{cr}}= (N+2)/(N-2)\), \(N\geq 3\). There are constructed radially symmetric sing-changing solutions, which blow-up at \(x = 0\) and \(t = T<\infty\), for \(N=3,4,5,6.\) These solutions exhibit fast blow-up; i.e. they satisfy NEWLINE\[NEWLINE \lim_{t\to T}(T-t)^{\frac{1}{p-1}} u(0,t) = \infty. NEWLINE\]NEWLINE In contrast, it is proved additionally, that radial positive decreasing solutions exhibit slow blow-up as in the subcritical case, which was discovered by Y. Giga and R. V. Kohn. This last result exists in the case of exponential nonlinearity and \(N=2\).