-
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
Blowup rate of solutions of a degenerate nonlinear parabolic equation
Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan |
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.
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. |
[2] |
B. Andrews,
Evolving convex curves, Calc. Var. & P.D.E., 7 (1998), 315-371.
doi: 10.1007/s005260050111. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. H. Tsai,
Blowup behavior of an equation arising from plane curves expansion, Diff. and Integ. Eq., 17 (2005), 849-872.
|
[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. |
[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. |
[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. |
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. |
[2] |
B. Andrews,
Evolving convex curves, Calc. Var. & P.D.E., 7 (1998), 315-371.
doi: 10.1007/s005260050111. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
D. H. Tsai,
Blowup behavior of an equation arising from plane curves expansion, Diff. and Integ. Eq., 17 (2005), 849-872.
|
[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. |
[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. |
[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. |
[1] |
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 |
[2] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[3] |
M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 |
[4] |
Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 |
[5] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[6] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[7] |
Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 |
[8] |
Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 |
[9] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[10] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[11] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[12] |
Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186 |
[13] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[14] |
Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 |
[15] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 |
[16] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
[17] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[18] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[19] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[20] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]