-
Previous Article
An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent
- DCDS-S Home
- This Issue
-
Next Article
A Liouville-type theorem for some Weingarten hypersurfaces
Singular backward self-similar solutions of a semilinear parabolic equation
1. | Mathematical Institute, Tohoku University, Sendai 980-8578, Japan |
2. | Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan |
References:
[1] |
C. J. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82 (1989), 207-218.
doi: doi:10.1016/0022-0396(89)90131-9. |
[2] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
[3] |
C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geometric Analysis, 9 (1999), 221-246. |
[4] |
Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math., 42 (1989), 845-884.
doi: doi:10.1002/cpa.3160420607. |
[5] |
L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226-1234; English translation: Differential Equation, 24 (1988), 799-805. |
[6] |
L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74, (in Russian). |
[7] |
N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal. 257 (2009), 2911-2937.
doi: doi:10.1016/j.jfa.2009.07.009. |
[8] |
N. Mizoguchi, On backward self-similar blowup solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 821-831.
doi: doi:10.1017/S0308210509000444. |
[9] |
Y. Naito and T. Suzuki, Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity, J. Differential Equations, 232 (2007), 176-211.
doi: doi:10.1016/j.jde.2006.07.012. |
[10] |
S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748.
doi: doi:10.1016/j.jde.2008.09.004. |
[11] |
S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Disc. Cont. Dyn. Systems, 26 (2010), 313-331. |
[12] |
S. Sato and E. Yanagida, Backward self-similar solution with a moving singularity for a semilinear parabolic equation, preprint. |
[13] |
T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396.
doi: doi:10.1512/iumj.2008.57.3269. |
[14] |
W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336.
doi: doi:10.1137/0518026. |
[15] |
L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996. |
show all references
References:
[1] |
C. J. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Differential Equations, 82 (1989), 207-218.
doi: doi:10.1016/0022-0396(89)90131-9. |
[2] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
[3] |
C.-C. Chen and C.-S. Lin, Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations, J. Geometric Analysis, 9 (1999), 221-246. |
[4] |
Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equation, Comm. Pure Appl. Math., 42 (1989), 845-884.
doi: doi:10.1002/cpa.3160420607. |
[5] |
L. A. Lepin, Countable spectrum of eigenfunctions of a nonlinear heat-conduction equation with distributed parameters, Differentsial'nye Uravneniya, 24 (1988), 1226-1234; English translation: Differential Equation, 24 (1988), 799-805. |
[6] |
L. A. Lepin, Self-similar solutions of a semilinear heat equation, Mat. Model., 2 (1990), 63-74, (in Russian). |
[7] |
N. Mizoguchi, Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation, J. Funct. Anal. 257 (2009), 2911-2937.
doi: doi:10.1016/j.jfa.2009.07.009. |
[8] |
N. Mizoguchi, On backward self-similar blowup solutions to a supercritical semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 821-831.
doi: doi:10.1017/S0308210509000444. |
[9] |
Y. Naito and T. Suzuki, Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity, J. Differential Equations, 232 (2007), 176-211.
doi: doi:10.1016/j.jde.2006.07.012. |
[10] |
S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748.
doi: doi:10.1016/j.jde.2008.09.004. |
[11] |
S. Sato and E. Yanagida, Forward self-similar solution with a moving singularity for a semilinear parabolic equation, Disc. Cont. Dyn. Systems, 26 (2010), 313-331. |
[12] |
S. Sato and E. Yanagida, Backward self-similar solution with a moving singularity for a semilinear parabolic equation, preprint. |
[13] |
T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J., 57 (2008), 3365-3396.
doi: doi:10.1512/iumj.2008.57.3269. |
[14] |
W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal., 18 (1987), 332-336.
doi: doi:10.1137/0518026. |
[15] |
L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations," Pitman Research Notes in Mathematics Series, 353, Longman, Harlow, 1996. |
[1] |
Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313 |
[2] |
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703 |
[3] |
Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857 |
[4] |
Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 |
[5] |
Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101 |
[6] |
Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108 |
[7] |
Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471 |
[8] |
Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002 |
[9] |
Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 |
[10] |
Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003 |
[11] |
Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 |
[12] |
Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 |
[13] |
Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323 |
[14] |
Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198 |
[15] |
Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks and Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401 |
[16] |
Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036 |
[17] |
D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685 |
[18] |
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131 |
[19] |
F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91 |
[20] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]