American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Numerical solution of partial differential equations with stochastic Neumann boundary conditions
  • DCDS-B Home
  • This Issue
  • Next Article
    $ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity
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  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

[1]

Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767
[2]

Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487
[3]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[4]

Pengyu Chen, Xuping Zhang, Yongxiang Li. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1975-1992. doi: 10.3934/cpaa.2018094
[5]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927
[6]

Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure & Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269
[7]

Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[8]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617
[9]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309
[10]

Xuan Liu, Ting Zhang. $ H^2 $ blowup result for a Schrödinger equation with nonlinear source term. Electronic Research Archive, 2020, 28 (2) : 777-794. doi: 10.3934/era.2020039
[11]

Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
[12]

Alexandre Montaru. Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 231-256. doi: 10.3934/dcdsb.2014.19.231
[13]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226
[14]

Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure & Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587
[15]

Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
[16]

Andrei Fursikov. The simplest semilinear parabolic equation of normal type. Mathematical Control & Related Fields, 2012, 2 (2) : 141-170. doi: 10.3934/mcrf.2012.2.141
[17]

Huashui Zhan. On a hyperbolic-parabolic mixed type equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 605-624. doi: 10.3934/dcdss.2017030
[18]

Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002
[19]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[20]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (132)
  • HTML views (353)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences