Abstract
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation
where p > 1; β > 0, q ≥ 1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antontsev S, Díaz J I, Shmarev S. Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, vol. 48. Boston: Birkäuser, 2002
Antontsev S, Shmarev S. Anisotropic parabolic equations with variable nonlinearity. Publ Mat, 2009, 53: 355–399
Antontsev S, Shmarev S. Energy solutions of evolution equations with nonstandard growth conditions. Monogr Real Acad Ci Exact Fís-Quím Nat Zaragoza, 2012, 38: 85–111
Attouchi A. Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion. J Differential Equations, 2012, 253: 2474–2492
Díaz J I. Qualitative study of nonlinear parabolic equations: An introduction. Extracta Math, 2001, 16: 303–341
DiBenedetto E. Degenerate Parabolic Equations. New York: Springer-Verlag, 1993
Fang Z B, Li G. Extinction and decay estimates of solutions for a class of doubly degenerate equations. Appl Math Lett, 2012, 25: 1795–1802
Fang Z B, Wang M, Li G. Extinction properties of solutions for a p-Laplacian evolution equation with nonlinear source and strong absorption. Math Aeterna, 2013, 3: 579–591
Fang Z B, Xu X H. Extinction behavior of solutions for the p-Laplacian equations with nonlocal sources. Nonlinear Anal Real World Appl, 2012, 13: 1780–1789
Fujita H. On the blowing up of solutions of the Cauchy problem for u t = Δu + u1+α. J Fac Sci Univ Tokyo Sect A Math, 1966, 16: 105–113
Galaktionov V A, Posashkov S A. Single point blow-up for N-dimensional quasilinear equations with gradient diffusion and source. Indiana Univ Math J, 1991, 40: 1041–1060
Galaktionov V A, Vázquez J L. Continuation of blowup solutions of nonlinear heat equations in several space dimensions. Comm Pure Appl Math, 1997, 50: 1–67
Giacomoni J, Sauvy P, Shmarev S. Complete quenching for a quasilinear parabolic equation. J Math Anal Appl, 2014, 410: 607–624
Gu Y G. Necessary and sufficient conditions of extinction of solution on parabolic equations. Acta Math Sin (Engl Ser), 1994, 37: 73–79
Hesaaraki M, Moameni A. Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in ℝN. Michigan Math J, 2004, 52: 375–389
Jin C H, Yin J X, Zheng S N. Critical Fujita absorption exponent for evolution p-Laplacian with inner absorption and boundary flux. Differential Integral Equations, 2014, 27: 643–658
Kwong Y C. Boundary behavior of the fast diffusion equation. Trans Amer Math Soc, 1990, 322: 263–283
Levine H A, Payne L E. Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations. J Math Anal Appl, 1976, 55: 329–334
Li Y X, Xie C H. Blow-up for p-Laplacian parabolic equations. Electron J Differential Equations, 2003, 20: 1–12
Lindqvist P. Notes on the p-Laplace equation. Http://www.math.ntnu.no/~lqvist/p-laplace.pdf, 2006
Ly I. The first eigenvalue for the p-Laplacian operator. JIPAM J Inequal Pure Appl Math, 2005, 6: Article 91
Mu C L, Zeng R. Single-point blow-up for a doubly degenerate parabolic equation with nonlinear source. Proc Roy Soc Edinburgh Sect A, 2011, 141: 641–654
Qu C Y, Bai X L, Zheng S N. Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions. J Math Anal Appl, 2014, 412: 326–333
Quittner P. Blow-up for semilinear parabolic equations with a gradient term. Math Methods Appl Sci, 1991, 14: 413–417
Quittner P, Souplet P. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Basel: Birkhäuser, 2007
Simon J. Compact sets in the space L p(0;T;B). Ann Mat Pura Appl (4), 1987, 146: 65–96
Souplet P, Weissler F B. Self-similar subsolutions and blowup for nonlinear parabolic equations. J Math Anal Appl, 1997, 212: 60–74
Vázquez J L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type. Oxford: Oxford University Press, 2006
Wang C P, Zheng S N, Wang Z J. Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data. Nonlinearity, 2007, 20: 1343–1359
Winkler M. A strongly degenerate diffusion equation with strong absorption. Math Nachr, 2004, 227: 83–101
Yang J G, Yang C X, Zheng S N. Second critical exponent for evolution p-Laplacian equation with weighted source. Math Comput Modelling, 2012, 56: 247–256
Yin J X, Jin C H. Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math Methods Appl Sci, 2007, 30: 1147–1167
Zhang Z C, Li Y. Blowup and existence of global solutions to nonlinear parabolic equations with degenerate diffusion. Electron J Differential Equations, 2013, 264: 1–17
Zhang Z C, Li Y. Classification of blowup solutions for a parabolic p-Laplacian equation with nonlinear gradient terms. J Math Anal Appl, 2016, 436: 1266–1283
Zhao J N. Existence and nonexistence of solutions for u t = div(|∇u|p-2∇u)+f(∇u; u; x; t). J Math Anal Appl, 1993, 172: 130–146
Zhao J N, Liang Z L. Blow-up rate of solutions for p-Laplacian equation. J Partial Differential Equations, 2008, 21: 134–140
Zhou J. Global existence and blow-up of solutions for a non-Newton polytropic filtration system with special volumetric moisture content. Comput Math Appl, 2016, 71: 1163–1172
Zhou J, Yang D. Upper bound estimate for the blow-up time of an evolution m-Laplace equation involving variable source and positive initial energy. Comput Math Appl, 2015, 69: 1463–1469
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11371286 and 11401458), the Special Fund of Education Department (Grant No. 2013JK0586) and the Youth Natural Science Grant of Shaanxi Province of China (Grant No. 2013JQ1015).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Y., Zhang, Z. & Zhu, L. Classification of certain qualitative properties of solutions for the quasilinear parabolic equations. Sci. China Math. 61, 855–868 (2018). https://doi.org/10.1007/s11425-016-9077-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9077-8