*(English)*Zbl 0555.35061

Since the pioneering work of *H. Fujita* [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.340)] the non-existence of global solutions to semilinear parabolic equations raised considerable interest [among the main results in this direction, recall *J. M. Ball*, Q. J. Math., Oxford II. Ser. 28, 473-486 (1977; Zbl 0377.35037), and of the author, Isr. J. Math. 38, 29-40 (1981; Zbl 0476.35043)]. In the present, stimulating paper, the author addresses the equation

It is well-known that for every $\phi \in {C}_{0}\left([-R,R]\right)$ there is a unique solution to (*) defined in a maximal time interval [0,T). It is also known that if $\phi $ is positive and ”large” enough, then $T<\infty $, moreover, as the author showed in the paper quoted above, if $p\ge 1$ and $p>(\gamma -1)/2,$ then ${\parallel u\left(t\right)\parallel}_{p}\to \infty $ as $t\to T\xb7$

Here the author considers initial data of the form $\phi =k\psi $, where $\psi $ is a positive solution of the associated stationary problem, and $k>1$ is chosen so large that the associated existence time is finite. He then proves that if $\gamma >2$ and is ”large”, then, as t approaches T, both u(t,x) and ${u}_{x}(t,x)$ have a finite limit for all $x=0:$ in other words, blow-up occurs only at the point $x=0$. The proof requires a number of steps, carried out in detail in a terse and comprehensive way. Since a similar single point blow-up behaviour prevails, too, for the purely reactive problem ${u}_{t}(t,x)={u}^{\gamma}(t,x)$ in $(-R,R)\times {\mathbb{R}}^{+},$ the author asks whether other properties of the latter problem are shared by (*), such as the boundedness of the p-norm of the solution as $t\to T$ for $1\le p\le (\gamma -1)/2,$ and the approximate representation of the solution $u(T,x)\sim {C\left|x\right|}^{-2(\gamma -1)}$ for X near 0. These items represent so far open, challenging questions. (Another open question is how far the single point blow-up depends on the fact that the positive initial function has itself a maximum at zero: had it two maxima, at, say, $\pm R/r$ $(r>1)$, would single point blow-up still occurr ?)