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Single point blow-up for a semilinear initial value problem. (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

$\left(*\right)\phantom{\rule{1.em}{0ex}}{u}_{t}\left(t,x\right)={u}_{xx}\left(t,x\right)+{u}^{\gamma }\left(t,x\right)\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}\left(-R,R\right)×{ℝ}^{+},\phantom{\rule{1.em}{0ex}}u\left(r,-R\right)=u\left(t,R\right)=0,\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{1.em}{0ex}}u\left(0,x\right)=\phi \left(x\right)·$

It is well-known that for every $\phi \in {C}_{0}\left(\left[-R,R\right]\right)$ 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 $p>\left(\gamma -1\right)/2,$ then ${\parallel u\left(t\right)\parallel }_{p}\to \infty$ as $t\to T·$

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 ${u}_{x}\left(t,x\right)$ 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 ${u}_{t}\left(t,x\right)={u}^{\gamma }\left(t,x\right)$ in $\left(-R,R\right)×{ℝ}^{+},$ 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 $1\le p\le \left(\gamma -1\right)/2,$ and the approximate representation of the solution $u\left(T,x\right)\sim {C|x|}^{-2\left(\gamma -1\right)}$ 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, $±R/r$ $\left(r>1\right)$, would single point blow-up still occurr ?)

Reviewer: P.de Mottoni

##### MSC:
 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions of PDE 35B35 Stability of solutions of PDE 35B60 Continuation of solutions of PDE 35K20 Second order parabolic equations, initial boundary value problems