Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions
Arkhipova, Arina A.
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000), p. 693-718 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 2000-01-01
Classification:  35B60,  35D05,  35J65,  35K50,  35K55 
@article{119203,
     author = {Arina A. Arkhipova},
     title = {Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities  I. On the continuability of smooth solutions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {41},
     year = {2000},
     pages = {693-718},
     zbl = {1046.35047},
     mrnumber = {1800172},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119203}
}

Arkhipova, Arina A. Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities  I. On the continuability of smooth solutions. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 693-718. http://gdmltest.u-ga.fr/item/119203/

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. | MR 1668390 | Zbl 0953.35059

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). (1999) | MR 1746069

Struwe M. On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558-581. (1985) | MR 0826871 | Zbl 0595.58013

Ladyzhenskaja O.A.; Solonnikov V.A.; Uraltseva N.N. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Society, Providence, 1968.

Giaquinta M.; Modica G. Local existence for quasilinear parabolic systems under non- linear boundary conditions, Ann. Mat. Pura Appl. 149 (1987), 41-59. (1987) | MR 0932775

Da Prato G. Spazi ${\Cal L}^{p, \tau}(Ømega, \delta)$ e loro proprieta, Annali di Matem. LXIX (1965), 383-392. (1965) | Zbl 0145.16207

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. (1966) | MR 0213737

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. | MR 1428990 | Zbl 0872.35020

Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Stud. 105, Princeton Univ. Press, Princeton, N.J., 1983. | MR 0717034 | Zbl 0516.49003

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. (1991) | MR 1118218