| Title:
|
Numerical study on the blow-up rate to a quasilinear parabolic equation (English) |
| Author:
|
Anada, Koichi |
| Author:
|
Ishiwata, Tetsuya |
| Author:
|
Ushijima, Takeo |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
325-330 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider the blow-up solutions for a quasilinear parabolic partial differential equation $u_t = u^2(u_{xx}+u)$. We numerically investigate the blow-up rates of these solutions by using a numerical method which is recently proposed by the authors [3]. (English) |
| Keyword:
|
Blow-up rate, type II blow-up, numerical estimate, scale invariance, rescaling algorithm, curvature flow |
| MSC:
|
35B44 |
| MSC:
|
35K59 |
| MSC:
|
65M99 |
| . |
| Date available:
|
2019-09-27T08:17:58Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703052 |
| . |
| Reference:
|
[1] Anada, K., Fukuda, I., Tsutsumi, M.: Regional blow-up and decay of solutions to the Initial-Boundary value problem for $u_t = uu_{xx} − \gamma(u_x)^2 + ku^2$., Funkcialaj Ekvacioj, 39 (1996), pp. 363–387. MR 1433906 |
| Reference:
|
[2] Anada, K., Ishiwata, T.: Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation., J. Differential Equations, 262 (2017), pp. 181–271. MR 3567485, 10.1016/j.jde.2016.09.023 |
| Reference:
|
[3] Anada, K., Ishiwata, T., Ushijima, T.: A numerical method of estimating blow-up rates for nonlinear evolution equations by using rescaling algorithm., to appear in Japan J. Ind. Appl. Math. MR 3768236 |
| Reference:
|
[5] Andrews, B.: Singularities in crystalline curvature flows., Asian J. Math., 6 (2002), pp. 101–122. MR 1902649, 10.4310/AJM.2002.v6.n1.a6 |
| Reference:
|
[6] Angenent, S. B.: On the formation of singularities in the curve shortening flow., J. Diff. Geo. 33 (1991), pp. 601–633. MR 1100205, 10.4310/jdg/1214446558 |
| Reference:
|
[7] Angenent, S. B., Velázquez, J. J. L.: Asymptotic shape of cusp singularities in curve shortening., Duke Math. J., 77 (1995), pp. 71–110. MR 1317628, 10.1215/S0012-7094-95-07704-7 |
| Reference:
|
[8] Berger, M., Kohn, R. V.: A rescaling algorithm for the numerical calculation of blowing-up solutions., Cmmm. Pure Appl. Math., 41 (1988), pp. 841–863. MR 0948774, 10.1002/cpa.3160410606 |
| Reference:
|
[9] Friedman, A., McLeod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations., Arch. Rational Mech. Anal., 96 (1987), pp. 55–80. MR 0853975, 10.1007/BF00251413 |
| Reference:
|
[10] Ishiwata, T., Yazaki, S.: On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion., J. Comput. Appl. Math., 159 (2003), pp. 55–64. MR 2022315, 10.1016/S0377-0427(03)00556-9 |
| Reference:
|
[11] Watterson, P. A.: Force-free magnetic evolution in the reversed-field pinch., Thesis, Cambridge University (1985). |
| Reference:
|
[12] Winkler, M.: Blow-up in a degenerate parabolic equation., Indiana Univ. Math. J., 53 (2004), pp. 1415–1442. MR 2104284, 10.1512/iumj.2004.53.2451 |
| . |
