zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

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