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)
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. (English) Zbl 1072.35098

The paper deals with the Cauchy–Dirichlet problem for one-dimensional semilinear parabolic equations with a gradient nonlinearity of the form

u t =u xx +|u x | p ,t>0,0<x<1,u(t,0)=0,u(t,1)=M,t>0,u(0,x)=u 0 (x),0<x<1,

where p>2, M0 and u 0 X:={vC 1 [0,1]:v(0)=0,v(1)=M}·

The authors provide a complete classification of large time behaviour of the classical solutions u· Precisely, either the space derivative u x blows up in a finite time with u itself remaining bounded, or u is global and converges in C 1 -norm to the unique steady state. The main difficulty concerns the proof of C 1 -boundedness of all global solutions, and in avoinding it the authors compute a nontrivial Lyapunov functional by carrying out a method introduced by T. Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, they argue by contradiction by showing that any C 1 unbounded global solution should converge to a singular stationary solution, which does not exist.

35K60Nonlinear initial value problems for linear parabolic equations
35K65Parabolic equations of degenerate type
35B45A priori estimates for solutions of PDE