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 Mat. Sb., 1999, Volume 190, Number 3, Pages 129–160 (Mi msb398)

Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains

A. E. Shishkov, A. G. Shchelkov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: A new approach (not based on the techniques of barriers) to the study of asymptotic properties of the generalized solutions of parabolic initial boundary-value problems with finite-time blow-up of the boundary values is proposed. Precise conditions on the blow-up pattern are found that guarantee uniform localization of the solution for an arbitrary compactly supported initial function. The main result of the paper consists in obtaining precise sufficient conditions for the singular (or blow-up) set of an arbitrary solution to remain within the boundary of the domain.

DOI: https://doi.org/10.4213/sm398

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English version:
Sbornik: Mathematics, 1999, 190:3, 447–479

Bibliographic databases:

UDC: 517.9
MSC: Primary 35K55; Secondary 35K65

Citation: A. E. Shishkov, A. G. Shchelkov, “Blow-up boundary regimes for general quasilinear parabolic equations in multidimensional domains”, Mat. Sb., 190:3 (1999), 129–160; Sb. Math., 190:3 (1999), 447–479

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb398
• https://doi.org/10.4213/sm398
• http://mi.mathnet.ru/eng/msb/v190/i3/p129

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. B. H. Gilding, J. Goncerzewicz, “Localization of Solutions of Exterior Domain Problems for the Porous Media Equation with Radial Symmetry”, SIAM J Math Anal, 31:4 (2000), 862
2. A. E. Shishkov, “Localized Boundary Blow-up Regimes for General Quasilinear Divergent Parabolic Equations of Arbitrary Order”, Proc. Steklov Inst. Math., 236 (2002), 341–356
3. Galaktionov V.A., Shishkov A.E., “Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations”, Proc. Roy. Soc. Edinburgh Sect. A, 133:5 (2003), 1075–1119
4. Galaktionov V.A., Shishkov A.E., “Structure of boundary blow-up for higher-order quasilinear parabolic equations”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460:2051 (2004), 3299–3325
5. Galaktionov, VA, “Self-similar boundary blow-up for higher-order quasilinear parabolic equations”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 135 (2005), 1195
6. Shishkov, A, “Diffusion versus absorption in semilinear elliptic equations”, Journal of Mathematical Analysis and Applications, 352:1 (2009), 206
7. Degtyarev S.P., “On the Instantaneous Shrinking of the Support of a Solution to the Cauchy Problem for an Anisotropic Parabolic Equation”, Ukr. Math. J., 61:5 (2009), 747–763
8. Shishkov A., “Large Solutions of Parabolic Logistic Equation With Spatial and Temporal Degeneracies”, Discret. Contin. Dyn. Syst.-Ser. S, 10:4 (2017), 895–907
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