##
**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
\[
(*)\quad u_ t(t,x)=u_{xx}(t,x)+u^{\gamma}(t,x)\quad in\quad (-R,R)\times {\mathbb{R}}^+,\quad u(r,-R)=u(t,R)=0,\quad t>0,\quad u(0,x)=\phi (x).
\]
It is well-known that for every \(\phi \in C_ 0([-R,R])\) 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 \(p\geq 1\) and \(p>(\gamma -1)/2,\) then \(\| u(t)\|_ p\to \infty\) 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 \(u_ x(t,x)\) 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 \(1\leq p\leq (\gamma -1)/2,\) and the approximate representation of the solution \(u(T,x)\sim C| x|^{-2(\gamma - 1)}\) 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 ?)

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 \(1\leq p\leq (\gamma -1)/2,\) and the approximate representation of the solution \(u(T,x)\sim C| x|^{-2(\gamma - 1)}\) 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 ?)

Reviewer: P.de Mottoni

### MSC:

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B35 | Stability in context of PDEs |

35B60 | Continuation and prolongation of solutions to PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

Full Text:
DOI

### References:

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