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)
The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source. (English) Zbl 1195.35068
Summary: The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form $$\frac {\partial u}{\partial t} = div (\rho(x) u^{m-1} |Du|^{\lambda -1} Du) + u^p$$ is studied. Global in time existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of a solution are obtained in the case of global solvability. A sharp universal (i.e., independent of the initial function) estimate of a solution near the blow-up time is obtained.

MSC:
35B44Blow-up (PDE)
35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35K59Quasilinear parabolic equations
WorldCat.org
Full Text: DOI
References:
[1] Tedeev, A. F.: The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. anal. 86, No. 6, 755-782 (2007) · Zbl 1129.35044 · doi:10.1080/00036810701435711
[2] Martynenko, A. V.; Tedeev, A. F.: On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source, Comput. math. Math. phys. 48, 1145-1160 (2008)
[3] Martynenko, A. V.; Tedeev, A. F.: Regularity of solutions of degenerate parabolic equation with inhomogenious density, Umb 5, No. 1, 116-145 (2008)
[4] Kamin, S.; Kersner, R.: Disappearence of interfaces in finite time, Meccanica 28, 117-120 (1993) · Zbl 0786.76088 · doi:10.1007/BF01020323
[5] Tedeev, A. F.: Conditions for the time global existence and nonexistence of a compact support to the Cauchy problem for quasilinear parabolic equations, Sib. math. J. 45, No. 1, 155-164 (2004) · Zbl 1053.35071 · emis:journals/SMZ/2004/01/content1.htm
[6] Degtyarev, S. P.; Tedeev, A. F.: L1--L$\infty $estimates of solutions of the Cauchy problem for an anisotropic degenerate parabolic equation with double non-linearity and growing initial data, Mat. sb. 198, No. 5, 45-66 (2007) · Zbl 1160.35453 · doi:10.1070/SM2007v198n05ABEH003853
[7] Kamin, S.; Rosenau, P.: Propagation of thermal waves in an inhomogeneous medium, Comm. pure appl. Math. 34, 831-852 (1981) · Zbl 0458.35042 · doi:10.1002/cpa.3160340605
[8] Kamin, S.; Rosenau, P.: Nonlinear diffusion in a finite mass medium, Comm. pure appl. Math. 35, 113-127 (1982) · Zbl 0469.35060 · doi:10.1002/cpa.3160350106
[9] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’tseva, N. N.: Linear and quasilinear equations of parabolic type, (1967) · Zbl 0164.12302
[10] Martynenko, A. V.; Tedeev, A. F.: Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogenious density, Comput. math. Math. phys. 47, 238-248 (2007) · Zbl 1210.35156 · doi:10.1134/S096554250702008X · http://www.maik.rssi.ru./abstract/commat/7/commat2_7p238abs.htm
[11] Otto, F.: L1-contraction and uniqueness for quasilinear elliptic--parabolic problems, J. differential equations 131, No. 1, 20-38 (1996) · Zbl 0862.35078 · doi:10.1006/jdeq.1996.0155
[12] Reyes, G.; Vazquez, J. L.: A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media, J. euro. Math. soc. 8, 531-554 (2006) · Zbl 1162.35033
[13] Eidelman, S. D.; Kamin, S.: On stabilization of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asympt. anal. 45, 55-71 (2005) · Zbl 1093.35043
[14] Eidus, D.; Kamin, S.: The filtration equation in a class of functions decreasing at infinity, Proc. amer. Math. soc. 120, No. 3, 825-830 (1994) · Zbl 0791.35065 · doi:10.2307/2160476
[15] Galaktionov, V. A.; Kamin, S.; Kersner, R.; Vazquez, J. L.: Intermediate asymptotics for inhomogeneous nonlinear heat conduction, J. math. Sci. 120, No. 3, 1277-1294 (2004) · Zbl 1205.35146 · doi:10.1023/B:JOTH.0000016049.94192.aa
[16] Galaktionov, V. A.; King, J. R.: On the behaviour of blow-up interfaces for an inhomogeneous filtration equation, J. appl. Math. 57, 53-77 (1996) · Zbl 0869.35076 · doi:10.1093/imamat/57.1.53
[17] Guedda, M.; Hilhorst, D.; Peletier, M.: Disappearing interfaces in nonlinear diffusion, Adv. math. Sci. appl. 7, 695-710 (1997) · Zbl 0891.35071
[18] Bernis, F.: Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. ann. 279, 373-394 (1988) · Zbl 0609.35048 · doi:10.1007/BF01456275
[19] Alt, H. W.; Luckhaus, S.: Quasilinear elliptic--parabolic differential equations, Math. Z. 183, 311-341 (1983) · Zbl 0497.35049 · doi:10.1007/BF01176474
[20] Tsutsumi, M.: On solutions of some doubly nonlinear degenerate parabolic equations with absorption, J. math. Anal. appl. 132, 187-212 (1988) · Zbl 0681.35047 · doi:10.1016/0022-247X(88)90053-4
[21] Andreucci, D.; Dibenedetto, E.: On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources equations, Ann. sc. Norm. super. Pisa cl. Sci. 18, 363-441 (1991) · Zbl 0762.35052 · numdam:ASNSP_1991_4_18_3_363_0
[22] 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
[23] Andreucci, D.; Tedeev, A. F.: A Fujita type result for degenerate Neumann problem in domains with noncompact boundary, J. math. Anal. appl. 231, 543-567 (1999) · Zbl 0920.35079 · doi:10.1006/jmaa.1998.6253
[24] Samarsky, A. A.; Galaktionov, V. A.; Kurdumov, S. P.; Mikhaylov, A. M.: Blowup in problems for quasilinear parabolic problems, (1987)
[25] Andreucci, D.; Tedeev, A. F.: Universal bounds at the blow-up time for nonlinear parabolic equations, Adv. difference equ. 10, No. 1, 89-120 (2005) · Zbl 1122.35042
[26] Bonafede, S.; Skrypnic, I. I.: On holder continuity of solutions of doubly nonlinear parabolic equations with weight, Ukrainian math. J. 51, No. 7, 890-903 (1999) · Zbl 0937.35021 · doi:10.1007/BF02529552
[27] Caffarelli, L.; Kohn, R.; Nirenberg, L.: First order interpolation inequalities with weights, Compos. math. 53, 259-275 (1984) · Zbl 0563.46024 · numdam:CM_1984__53_3_259_0