| Title:
|
Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions (English) |
| Author:
|
Arkhipova, A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
41 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
693-718 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval $[0,T)$ solution to the Cauchy-Neumann problem is studied. For the situation when the ``local energies'' of the solution are uniformly bounded on $[0,T)$, smooth extendibility of the solution up to $t=T$ is proved. In the case when $[0,T)$ defines the maximal interval of the existence of a smooth solution, the singular set at the moment $t=T$ is described. (English) |
| Keyword:
|
boundary value problem |
| Keyword:
|
nonlinear parabolic systems |
| Keyword:
|
solvability |
| MSC:
|
35B60 |
| MSC:
|
35D05 |
| MSC:
|
35J65 |
| MSC:
|
35K50 |
| MSC:
|
35K55 |
| idZBL:
|
Zbl 1046.35047 |
| idMR:
|
MR1800172 |
| . |
| Date available:
|
2009-01-08T19:06:34Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119203 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/119223 |
| . |
| Reference:
|
[1] Arkhipova A.: Global solvability of the Cauchy-Dirichlet Problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables.Probl. Mat. Anal., no. 16, S.-Petersburg Univ., S.-Petersburg (1997), pp.3-40; English transl.: J. Math. Sci. 92 (1998), no. 6, 4231-4255. Zbl 0953.35059, MR 1668390 |
| Reference:
|
[2] Arkhipova A.: Local and global in time solvability of the Cauchy-Dirichlet problem to a class of nonlinear nondiagonal parabolic systems.Algebra & Analysis 11 6 (1999), 81-119 (Russian). MR 1746069 |
| Reference:
|
[3] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces.Comment. Math. Helv. 60 (1985), 558-581. Zbl 0595.58013, MR 0826871 |
| Reference:
|
[4] Ladyzhenskaja O.A., Solonnikov V.A., Uraltseva N.N.: Linear and Quasilinear Equations of Parabolic Type.Amer. Math. Society, Providence, 1968. |
| Reference:
|
[5] Giaquinta M., Modica G.: Local existence for quasilinear parabolic systems under non- linear boundary conditions.Ann. Mat. Pura Appl. 149 (1987), 41-59. MR 0932775 |
| Reference:
|
[6] Da Prato G.: Spazi ${\Cal L}^{p, \tau}(Ømega, \delta)$ e loro proprieta.Annali di Matem. LXIX (1965), 383-392. Zbl 0145.16207 |
| Reference:
|
[7] Campanato S.: Equazioni paraboliche del secondo ordine e spazi ${\Cal L}^{2, \delta} (Ømega, \delta)$..Ann. Mat. Pura Appl. 73 (1966), ser.4, 55-102. MR 0213737 |
| Reference:
|
[8] Arkhipova A.: On the Neumann problem for nonlinear elliptic systems with quadratic nonlinearity.St. Petersburg Math. J. 8 (1997), no. 5, 1-17; in Russian: Algebra & Analysis, St. Petersburg 8 (1996), no. 5. Zbl 0872.35020, MR 1428990 |
| Reference:
|
[9] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems.Ann. Math. Stud. 105, Princeton Univ. Press, Princeton, N.J., 1983. Zbl 0516.49003, MR 0717034 |
| Reference:
|
[10] Nečas J., Šverák V.: On regularity of solutions of nonlinear parabolic systems.Ann. Scuola Norm. Sup. Pisa 18 ser. IV, F.1 (1991), 1-11. MR 1118218 |
| . |
