no. 1
Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 153-167.

A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval [0,T) is studied. Smooth extendibility of the solution up to t=T is proved, provided that “normilized local energies” of the solution are uniformly bounded on [0,T). For the case where [0,T) determines the maximal interval of existence of a smooth solution,the Hausdorff measure of a singular set at the moment t=T is estimated.

Classification : 35K50, 35K45, 35K60 
@article{ASNSP_2002_5_1_1_153_0,
     author = {Arkhipova, Arina},
     title = {Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {153--167},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {1},
     year = {2002},
     mrnumber = {1994805},
     zbl = {1049.35093},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_1_153_0/}
}
TY  - JOUR
AU  - Arkhipova, Arina
TI  - Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 153
EP  - 167
VL  - 1
IS  - 1
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2002_5_1_1_153_0/
LA  - en
ID  - ASNSP_2002_5_1_1_153_0
ER  - 
%0 Journal Article
%A Arkhipova, Arina
%T Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 153-167
%V 1
%N 1
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2002_5_1_1_153_0/
%G en
%F ASNSP_2002_5_1_1_153_0
Arkhipova, Arina. Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 153-167. http://www.numdam.org/item/ASNSP_2002_5_1_1_153_0/

[1] H. Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), no. 2, 153-192. | MR | Zbl

[2] A. Arkhipova, Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables, J. Math. Sci. 92 (1998), no. 6, 4231-4255. | MR | Zbl

[3] A. Arkhipova, Local and global in time solvability of the Cauchy-Dirichlet problem to a class of nonlinear nondiagonal parabolic systems, St. Petersburg Math J. 11 (2000), no. 6, 81-119. | MR | Zbl

[4] A. Arkhipova, Cauchy-Neumann problem to a class of nondiagonal parabolic systems with quadratic growth nonlinearities. I. Local and global solvability results, Comment. Math. Univ. Carolin. 41, (2000), no. 4, 693-718. | MR | Zbl

[5] A. Arkhipova, Cauchy-Neumann problem to a class of nondiagonal parabolic systems with quadratic growth nonlinearities. II. Continuability of smooth solutions, to appear in Comment. Math. Univ. Carolin. (2000). | MR | Zbl

[6] S. Campanato, Equazioni paraboliche del secondo ordine e spazi ℒ 2,δ (Ω,δ), Ann. Mat. Pura Appl., Ser. 4 73 (1966), 55-102. | MR | Zbl

[7] M. Giaquinta, “Multiple integrals in the calculus of variations and nonlinear elliptic systems”, Princeton, NJ, 1983. | MR | Zbl

[8] M. Giaquinta - G. Modica, Local existence for quasilinear parabolic systems under non-linear boundary conditions, Ann. Mat. Pura Appl. Ser. 4, 149 (1987), 41-59. | MR | Zbl

[9] O. A. Ladyzhenskaja - V. A. Solonnikov - N. N. Uraltseva, “Linear and quasilinear equations of parabolic type”, Amer. Math. Soc. Providence, RI, 1968.