## 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 ?)
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

### Citations:

Zbl 0163.340; Zbl 0377.35037; Zbl 0476.35043
Full Text:

### References:

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