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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

(*)u t (t,x)=u xx (t,x)+u γ (t,x)in(-R,R)× + ,u(r,-R)=u(t,R)=0,t>0,u(0,x)=ϕ(x)·

It is well-known that for every ϕC 0 ([-R,R]) there is a unique solution to (*) defined in a maximal time interval [0,T). It is also known that if ϕ is positive and ”large” enough, then T<, moreover, as the author showed in the paper quoted above, if p1 and p>(γ-1)/2, then u(t) p as tT·

Here the author considers initial data of the form ϕ=kψ, where ψ 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 γ>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 γ (t,x) in (-R,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 tT for 1p(γ-1)/2, and the approximate representation of the solution u(T,x)C|x| -2(γ-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, ±R/r (r>1), would single point blow-up still occurr ?)

Reviewer: P.de Mottoni

MSC:
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