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October  2019, 24(10): 5317-5336. doi: 10.3934/dcdsb.2019060

Blowup rate of solutions of a degenerate nonlinear parabolic equation

Chi-Cheung Poon

Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan

Received  May 2018 Revised  September 2018 Published  October 2019 Early access  April 2019

We study a nonlinear parabolic equation arising from heat combustion and plane curve evolution problems. Suppose that a solution satisfies a symmetry condition and blows up of type Ⅱ. We give an upper bound and a lower bound for the blowup rate of the solution. The lower bound obtained here is probably optimal.

Keywords: Degenerate parabolic equation, nonlinear parabolic equation, plane curve evolution, type Ⅱ blowup, blowup rate.
Mathematics Subject Classification: 35K20, 35K55, 35K65.
Citation: Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060
References:
[1]

K. Anada and T. Ishiwata, Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation, J. Differential Equations, 262 (2017), 181-271.  doi: 10.1016/j.jde.2016.09.023.  Google Scholar

[2]

B. Andrews, Evolving convex curves, Calc. Var. & P.D.E., 7 (1998), 315-371.  doi: 10.1007/s005260050111.  Google Scholar

[3]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[4]

S. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom., 33 (1991), 601-633.  doi: 10.4310/jdg/1214446558.  Google Scholar

[5]

S. Angenent and J. Velazquez, Asymptotic shape of cusp singularities in curve shortening, Duke Math. Journal, 77 (1995), 71-110.  doi: 10.1215/S0012-7094-95-07704-7.  Google Scholar

[6]

A. Friedman and B. McLeod, Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. Rational Mech. Anal., 96 (1986), 55-80.  doi: 10.1007/BF00251413.  Google Scholar

[7]

T. C. Lin, C. C. Poon and D. H Tsai, Expanding convex immersed closed plane curves, Calc. Var. & P.D.E., 34 (2009), 153-178.  doi: 10.1007/s00526-008-0180-7.  Google Scholar

[8]

Y. C. Lin, C. C. Poon and D. H. Tsai, Contracting convex immersed closed plane curves with slow speed of curvature, Transactions of AMS, 364 (2012), 5735-5763.  doi: 10.1090/S0002-9947-2012-05611-X.  Google Scholar

[9]

C. C. Poon and D. H Tsai, Contracting convex immersed closed plane curves with fast speed of curvature, Comm. Anal. Geom., 18 (2010), 23-75.  doi: 10.4310/CAG.2010.v18.n1.a2.  Google Scholar

[10]

D. H. Tsai, Blowup behavior of an equation arising from plane curves expansion, Diff. and Integ. Eq., 17 (2005), 849-872.   Google Scholar

[11]

J. Urbas, Convex curves moving homotheticallt by negative powers of their curvature, Asian J. Math., 3 (1999), 635-656.  doi: 10.4310/AJM.1999.v3.n3.a4.  Google Scholar

[12]

M. Winkler, Blow-up of solutions to a degenerate parabolic equation not in divergence form, J. Differential Equation, 192 (2003), 445-474.  doi: 10.1016/S0022-0396(03)00127-X.  Google Scholar

[13]

M. Winkler, Blow-up in a degenerate parabolic equation, Indiana Univ. Math. Journal, 53 (2004), 1415-1442.  doi: 10.1512/iumj.2004.53.2451.  Google Scholar

show all references

References:
[1]

K. Anada and T. Ishiwata, Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation, J. Differential Equations, 262 (2017), 181-271.  doi: 10.1016/j.jde.2016.09.023.  Google Scholar

[2]

B. Andrews, Evolving convex curves, Calc. Var. & P.D.E., 7 (1998), 315-371.  doi: 10.1007/s005260050111.  Google Scholar

[3]

S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.  doi: 10.1515/crll.1988.390.79.  Google Scholar

[4]

S. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom., 33 (1991), 601-633.  doi: 10.4310/jdg/1214446558.  Google Scholar

[5]

S. Angenent and J. Velazquez, Asymptotic shape of cusp singularities in curve shortening, Duke Math. Journal, 77 (1995), 71-110.  doi: 10.1215/S0012-7094-95-07704-7.  Google Scholar

[6]

A. Friedman and B. McLeod, Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. Rational Mech. Anal., 96 (1986), 55-80.  doi: 10.1007/BF00251413.  Google Scholar

[7]

T. C. Lin, C. C. Poon and D. H Tsai, Expanding convex immersed closed plane curves, Calc. Var. & P.D.E., 34 (2009), 153-178.  doi: 10.1007/s00526-008-0180-7.  Google Scholar

[8]

Y. C. Lin, C. C. Poon and D. H. Tsai, Contracting convex immersed closed plane curves with slow speed of curvature, Transactions of AMS, 364 (2012), 5735-5763.  doi: 10.1090/S0002-9947-2012-05611-X.  Google Scholar

[9]

C. C. Poon and D. H Tsai, Contracting convex immersed closed plane curves with fast speed of curvature, Comm. Anal. Geom., 18 (2010), 23-75.  doi: 10.4310/CAG.2010.v18.n1.a2.  Google Scholar

[10]

D. H. Tsai, Blowup behavior of an equation arising from plane curves expansion, Diff. and Integ. Eq., 17 (2005), 849-872.   Google Scholar

[11]

J. Urbas, Convex curves moving homotheticallt by negative powers of their curvature, Asian J. Math., 3 (1999), 635-656.  doi: 10.4310/AJM.1999.v3.n3.a4.  Google Scholar

[12]

M. Winkler, Blow-up of solutions to a degenerate parabolic equation not in divergence form, J. Differential Equation, 192 (2003), 445-474.  doi: 10.1016/S0022-0396(03)00127-X.  Google Scholar

[13]

M. Winkler, Blow-up in a degenerate parabolic equation, Indiana Univ. Math. Journal, 53 (2004), 1415-1442.  doi: 10.1512/iumj.2004.53.2451.  Google Scholar

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