# 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 some nonlinear parabolic problems under mixed boundary conditions. (English) Zbl 1201.35057
Summary: We determine the lower bound for the blow-up time of solution to equations of the form $u_t = \text{div} (\rho (|\nabla u|^2)\operatorname{grad} u) + f(u)$ if the solution blows up. Conditions which ensure, that blow-up does not occur, are also presented.

##### MSC:
 35B44 Blow-up (PDE) 35K59 Quasilinear parabolic equations 35K20 Second order parabolic equations, initial boundary value problems
Full Text:
##### References:
 [1] Bandle, C.; Brunner, H.: Blow-up in diffusion equations: A survey. J. comput. Appl. math. 97, 3-22 (1998) · Zbl 0932.65098 [2] Galaktionov, V. A.; Vázquez, J. L.: The problem of blow up in nonlinear parabolic equations. Discrete contin. Dyn. syst. 8, 399-433 (2002) · Zbl 1010.35057 [3] Levine, H. A.: The role of critical exponents in blow-up theorems. SIAM rev. 32, 262-288 (1990) · Zbl 0706.35008 [4] Straughan, B.: Instability, nonexistence and weighted energy methods in fluid dynamics and related theories. Res. notes math. 74 (1982) · Zbl 0492.76001 [5] Ball, J. M.: Remarks on blow up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford 28, 473-486 (1977) · Zbl 0377.35037 [6] Caffarrelli, L. A.; Friedman, A.: Blow-up of solutions of nonlinear heat equations. J. math. Anal. appl. 129, 409-419 (1988) · Zbl 0653.35038 [7] Friedman, A.; Mcleod, B.: Blow-up of positive solutions of semilinear heat equations. Indiana univ. Math. J. 34, 425-447 (1985) · Zbl 0576.35068 [8] Kielhöfer, H.: Existen und regularität von lösungen semilinearer parabolisccher anfangs--randwertprobleme. Math. Z. 142, 131-160 (1975) · Zbl 0324.35047 [9] Levine, H. A.: Nonexietence 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 [10] 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 [11] Straughan, B.: Explosive instabilities in mechamics. (1998) · Zbl 0911.35002 [12] Payne, L. E.; Philippin, G. A.; Schaefer, P. W.: Blow-up phenomena for some nonlinear parabolic problems. Nonlinear anal. 69, 3495-3502 (2008) · Zbl 1159.35382 [13] Friedman, A.: Remarks on the maximum principle for parabolic equations and its applications. Pacific J. Math. 8, 201-211 (1958) · Zbl 0103.06403 [14] Nirenberg, L.: A strong maximum principle for parabolic equations. Commun. pure appl. Math. 6, 167-177 (1953) · Zbl 0050.09601