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)
Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values. (English) Zbl 1182.35028
Summary: We consider the positive solution of a Cauchy problem for the following $p$-Laplace parabolic equation $$u_t = \text{div}(|\nabla u|^{p-2}\nabla u)+u^q,\quad p>2,\ q>1,$$ and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence of global and non-global solutions of the Cauchy problem. Furthermore, the life span of solutions is also studied.

35B33Critical exponents (PDE)
35K65Parabolic equations of degenerate type
35B44Blow-up (PDE)
35K92Quasilinear parabolic equations with $p$-Laplacian
35K15Second order parabolic equations, initial value problems
Full Text: DOI
[1] Alikakos, N. D.; Evans, L. C.: Continuity of the gradient for weak solutions of degenerate parabolic equation, J. math. Pures appl. 62, 253-268 (1983) · Zbl 0529.35039
[2] Deng, K.; Levine, H. A.: The role of critical exponents in blow-up theorems: the sequel, J. math. Anal. appl. 243, 85-126 (2000) · Zbl 0942.35025 · doi:10.1006/jmaa.1999.6663
[3] Dibenedetto, E.; Friedman, A.: Hölder estimates for nonlinear degenerate parabolic system, J. reine angew. Math. 357, 1-22 (1985) · Zbl 0549.35061 · doi:10.1515/crll.1985.357.1 · crelle:GDZPPN002202239
[4] Dibenedetto, E.; Herrero, M. A.: On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. amer. Math. soc. 314, 187-224 (1989) · Zbl 0691.35047 · doi:10.2307/2001442
[5] Friedman, A.; Mcleod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. ration. Mech. anal. 96, 55-80 (1987) · Zbl 0619.35060
[6] Fujita, H.: On the blowing up of solutions of the Cauchy problem for ut=?$u+u1+{\alpha}$, J. fac. Sci. univ. Tokyo sec. A 16, 105-113 (1966)
[7] Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P.; Samarskii, A. A.: Unbounded solutions of the Cauchy problem for the parabolic equation $ut=\nabla (u{\alpha}\nabla u)+u{\beta}$, Soviet phys. Dokl. 25, 458-459 (1980) · Zbl 0515.35045
[8] Galaktionov, V. A.: Conditions for nonexistence as a whole and localization of the solutions of Cauchy’s problem for a class of nonlinear parabolic equations, Zh. vychisl. Mat. mat. Fiz. 23, 1341-1354 (1985) · Zbl 0576.35055 · doi:10.1016/S0041-5553(83)80073-1
[9] Galaktionov, V. A.: Blow-up for quasilinear heat equations with critical Fujita’s exponents, Proc. roy. Soc. Edinburgh 124A, 517-525 (1994) · Zbl 0808.35053 · doi:10.1017/S0308210500028766
[10] Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P.; Samarskii, A. A.: Blow-up in quasilinear parabolic equations, (1995) · Zbl 1020.35001
[11] Galaktionov, V. A.; Levine, H. A.: A general approach to critical Fujita exponents and systems, Nonlinear anal. TMA 34, 1005-1027 (1998) · Zbl 1139.35317 · doi:10.1016/S0362-546X(97)00716-5
[12] Gui, C. F.; Wang, X. F.: Life span of solutions of the Cauchy problem for a semilinear heat equation, J. differential equations 115, 166-172 (1995) · Zbl 0813.35034 · doi:10.1006/jdeq.1995.1010
[13] Guo, J. S.: Similarity solutions for a quasilinear parabolic equation, J. austral. Math. soc. Ser. B 37, 253-266 (1995) · Zbl 0859.35063 · doi:10.1017/S0334270000007694
[14] Guo, J. S.; Guo, Y. Y.: On a fast diffusion equation with source, Tohoku math. J. 53, 571-579 (2001) · Zbl 0995.35035 · doi:10.2748/tmj/1113247801
[15] Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic equation, Proc. Japan acad. 49, 503-505 (1973) · Zbl 0281.35039 · doi:10.3792/pja/1195519254
[16] Huang, Q.; Mochizuki, K.; Mukai, K.: Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values, Hokkaido math. J. 27, 393-407 (1998) · Zbl 0906.35044
[17] Lee, T. Y.; Ni, W. M.: Global existence, large time behavior and life span on solutions of a semilinear parabolic Cauchy problem, Trans. amer. Math. soc. 333, 365-378 (1992) · Zbl 0785.35011 · doi:10.2307/2154114
[18] Levine, H. A.: The role of critical exponents in blowup theorems, SIAM rev. 32, 262-288 (1990) · Zbl 0706.35008 · doi:10.1137/1032046
[19] Li, Y. H.; Mu, C. L.: Life span and a new critical exponent for a degenerate parabolic equation, J. differential equations 207, 392-406 (2004) · Zbl 1066.35047 · doi:10.1016/j.jde.2004.08.024
[20] Luckhaus, S.; Dal Passo, R.: A degenerate diffusion problem not in divergence form, J. differential equations 69, 1-14 (1987) · Zbl 0688.35045 · doi:10.1016/0022-0396(87)90099-4
[21] Mochizuki, K.; Mukai, K.: Existence and nonexistence of global solutions to fast diffusions with source, Methods appl. Anal. 2, 92-102 (1995) · Zbl 0832.35083
[22] Mochizuki, K.; Suzuki, R.: Critical exponent and critical blow-up for quasilinear parabolic equations, Irsael. J. Math. 98, 141-156 (1997) · Zbl 0880.35057 · doi:10.1007/BF02937331
[23] Mukai, K.; Mochizuki, K.; Huang, Q.: Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values, Nonlinear anal. TMA 39, 33-45 (2000) · Zbl 0936.35034 · doi:10.1016/S0362-546X(98)00161-8
[24] Qi, Y. W.; Levine, H. A.: The critical exponent of degenerate parabolic systems, Z. angew. Math. phys. 44, 249-265 (1993) · Zbl 0816.35068 · doi:10.1007/BF00914283
[25] Qi, Y. W.: Critical exponents of degenerate parabolic equations, Sci. China 38A, 1153-1162 (1995) · Zbl 0837.35076
[26] Qi, Y. W.: The global existence and nonuniqueness of a nonlinear degenerate equations, Nonlinear anal. TMA 31, 117-136 (1998) · Zbl 0907.35073 · doi:10.1016/S0362-546X(96)00298-2
[27] Qi, Y. W.: The critical exponents of parabolic equations and blow-up in RN, Proc. roy. Soc. Edinburgh 128A, 123-136 (1998) · Zbl 0892.35088 · doi:10.1017/S0308210500027190
[28] Weissler, F. B.: Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38, 29-40 (1981) · Zbl 0476.35043 · doi:10.1007/BF02761845
[29] Wiegner, M.: A degenerate diffusion equation with a nonlinear source term, Nonlinear anal. TMA 28, 1977-1995 (1997) · Zbl 0874.35061 · doi:10.1016/S0362-546X(96)00027-2
[30] Wiegner, M.: Blow-up for solutions of some degenerate parabolic equations, Differential integral equations 7, 1641-1647 (1994) · Zbl 0797.35100
[31] M. Winkler, Some results on degenerate parabolic equations not in divergence form, Ph.D. Thesis, Aachen, 2000
[32] Winkler, M.: On the Cauchy problem for a degenerate parabolic equation, Z. anal. Anwendungen 20, 677-690 (2001) · Zbl 0987.35089 · http://www.heldermann.de/zaaabs20.htm
[33] Winkler, M.: A critical exponent in a degenerate parabolic equation, Math. meth. Appl. sci. 25, 911-925 (2002) · Zbl 1007.35043 · doi:10.1002/mma.319
[34] Zhao, J. N.: The asymptotic behavior of solutions of a quasilinear degenerate parabolic equation, J. differential equations 102, 33-52 (1993) · Zbl 0816.35070 · doi:10.1006/jdeq.1993.1020
[35] Zhao, J. N.: Source type solutions of a quasilinear degenerate parabolic with absorption, Chin. ann. Math. 15B, 89-104 (1994) · Zbl 0792.35105
[36] Zhao, J. N.: On the Cauchy problem and initial traces for the evolution P-equation with strongly nonlinear sources, J. differential equations 121, 329-383 (1995) · Zbl 0836.35081 · doi:10.1006/jdeq.1995.1132