Weissler, Fred B. Single point blow-up for a semilinear initial value problem. (English) Zbl 0555.35061 J. Differ. Equations 55, 204-224 (1984). 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 ?) Reviewer: P.de Mottoni Cited in 1 ReviewCited in 86 Documents 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 Keywords:non-existence; global solutions; semilinear parabolic equations; initial data; blow-up Citations:Zbl 0163.340; Zbl 0377.35037; Zbl 0476.35043 PDFBibTeX XMLCite \textit{F. B. Weissler}, J. Differ. Equations 55, 204--224 (1984; Zbl 0555.35061) Full Text: DOI References: [1] Ball, J. M., Remarks on blow-up and nonexistence theorems for nonlinear evoluation equations, Quart. J. Math. Oxford Ser., 28, 473-486 (1977) · Zbl 0377.35037 [2] Ball, J. M., Finite time blow-up in nonlinear problems, (Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin. 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Mech. and PDE (1978), North-Holland: North-Holland Amsterdam), 452-465 [18] Strauss, W. A., Everywhere defined wave operators, (Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin. Proceedings, Symposium on Nonlinear Evolution Equations, University of Wisconsin, Madison, October 1977 (1978), Academic Press: Academic Press New York) · Zbl 0466.47005 [19] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J., 29, 79-102 (1980) · Zbl 0443.35034 [20] Weissler, F. B., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38, 29-40 (1981) · Zbl 0476.35043 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.