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Single point blow-up for a semilinear initial value problem. (English) Zbl 0555.35061
Since the pioneering work of {\it 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 {\it 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 $$ (*)\quad u\sb t(t,x)=u\sb{xx}(t,x)+u\sp{\gamma}(t,x)\quad in\quad (-R,R)\times {\bbfR}\sp+,\quad u(r,-R)=u(t,R)=0,\quad t>0,\quad u(0,x)=\phi (x). $$ It is well-known that for every $\phi \in C\sb 0([-R,R])$ 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 $p\ge 1$ and $p>(\gamma -1)/2,$ then $\Vert u(t)\Vert\sb p\to \infty$ 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 $u\sb x(t,x)$ 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 $u\sb t(t,x)=u\sp{\gamma}(t,x)$ in $(-R,R)\times {\bbfR}\sp+,$ 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 $u(T,x)\sim C\vert x\vert\sp{-2(\gamma - 1)}$ 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

35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35B35Stability of solutions of PDE
35B60Continuation of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
Full Text: DOI
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