zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition. (English) Zbl 1206.35151
Summary: This note deals with a class of heat emission processes in a medium with a non-negative source, a nonlinear decreasing thermal conductivity and a linear radiation (Robin) boundary condition. For such heat emission problems, we make use of a first-order differential inequality technique to establish conditions on the data sufficient to guarantee that the blow-up of the solutions does occur or does not occur. In addition, the same technique is used to determine a lower bound for the blow-up time when blow-up occurs.

MSC:
35K59Quasilinear parabolic equations
35K20Second order parabolic equations, initial boundary value problems
35B44Blow-up (PDE)
WorldCat.org
Full Text: DOI
References:
[1] Quittner, R.; Souplet, P.: Superlinear parabolic problems. Blow-up, global existence and steady states, (2007) · Zbl 1128.35003
[2] Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P.: Blow-up in quasilinear parabolic equations, De gruyter expositions in mathematics 19 (1995) · Zbl 1020.35001
[3] Galaktionov, V. A.; Vazquez, J. L.: The problem of blow-up in nonlinear parabolic equations, Discrete contin. Dyn. syst. 8, 399-433 (2002) · Zbl 1010.35057 · doi:10.3934/dcds.2002.8.399
[4] Levine, H. A.: Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients, Math. ann. 214, 205-220 (1975) · Zbl 0286.35006 · doi:10.1007/BF01352106
[5] Payne, L. E.; Schaefer, P. W.: Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. math. Anal. appl. 328, 1196-1205 (2007) · Zbl 1110.35031 · doi:10.1016/j.jmaa.2006.06.015
[6] Payne, L. E.; Schaefer, P. W.: Blow-up in parabolic problems under Robin boundary conditions, Appl. anal. 87, 699-707 (2008) · Zbl 1156.35408 · doi:10.1080/00036810802189662
[7] Payne, L. E.; Schaefer, P. W.: Bounds for the blow-up time for the heat equation under nonlinear boundary conditions, Proc. roy. Soc. Edinburgh sect. A 139, 1289-1296 (2009) · Zbl 1184.35077 · doi:10.1017/S0308210508000802
[8] Payne, L. E.; Schaefer, P. W.: Lower bounds for blow-up time in parabolic problems under Neumann conditions, Appl. anal. 85, 1301-1311 (2006) · Zbl 1110.35032 · doi:10.1080/00036810600915730
[9] Enache, C.: Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear anal. 69, 2864-2874 (2008) · Zbl 1158.35375 · doi:10.1016/j.na.2007.08.063
[10] Daners, D.: A Faber--krahn inequality for Robin problems in any space dimension, Math. ann. 335, 767-785 (2006) · Zbl 1220.35103 · doi:10.1007/s00208-006-0753-8