Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions

Arina A. Arkhipova

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 4, page 693-718
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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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 41.4 (2000): 693-718. <http://eudml.org/doc/248601>.

@article{Arkhipova2000,
abstract = {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.},
author = {Arkhipova, Arina A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {boundary value problem; nonlinear parabolic systems; solvability; boundary value problem; nonlinear parabolic system},
language = {eng},
number = {4},
pages = {693-718},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions},
url = {http://eudml.org/doc/248601},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Arkhipova, Arina A.
TI - Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 4
SP - 693
EP - 718
AB - 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.
LA - eng
KW - boundary value problem; nonlinear parabolic systems; solvability; boundary value problem; nonlinear parabolic system
UR - http://eudml.org/doc/248601
ER -

References

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  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. Zbl0953.35059MR1668390
  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). (1999) MR1746069
  3. Struwe M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558-581. (1985) Zbl0595.58013MR0826871
  4. Ladyzhenskaja O.A., Solonnikov V.A., Uraltseva N.N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Society, Providence, 1968. 
  5. Giaquinta M., Modica G., Local existence for quasilinear parabolic systems under non- linear boundary conditions, Ann. Mat. Pura Appl. 149 (1987), 41-59. (1987) MR0932775
  6. Da Prato G., Spazi ℒ p , τ ( Ø m e g a , δ ) e loro proprieta, Annali di Matem. LXIX (1965), 383-392. (1965) Zbl0145.16207
  7. Campanato S., Equazioni paraboliche del secondo ordine e spazi ℒ 2 , δ ( Ø m e g a , δ ) ., Ann. Mat. Pura Appl. 73 (1966), ser.4, 55-102. (1966) MR0213737
  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. Zbl0872.35020MR1428990
  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. Zbl0516.49003MR0717034
  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. (1991) MR1118218

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