Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems

Arina Arkhipova

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 153-167
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Arkhipova, Arina. "Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 153-167. <http://eudml.org/doc/84461>.

@article{Arkhipova2002,
abstract = {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.},
author = {Arkhipova, Arina},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Euler-Lagrange equations; Morrey spaces; Campanato spaces; maximal solutions},
language = {eng},
number = {1},
pages = {153-167},
publisher = {Scuola normale superiore},
title = {Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems},
url = {http://eudml.org/doc/84461},
volume = {1},
year = {2002},
}

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
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 153
EP - 167
AB - 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.
LA - eng
KW - Euler-Lagrange equations; Morrey spaces; Campanato spaces; maximal solutions
UR - http://eudml.org/doc/84461
ER -

References

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  1. [1] H. Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), no. 2, 153-192. Zbl0596.35061MR816618
  2. [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. Zbl0953.35059MR1668390
  3. [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. Zbl0973.35095MR1746069
  4. [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. Zbl1046.35047MR1800172
  5. [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). Zbl1046.35047MR1800172
  6. [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. Zbl0144.14101MR213737
  7. [7] M. Giaquinta, “Multiple integrals in the calculus of variations and nonlinear elliptic systems”, Princeton, NJ, 1983. Zbl0516.49003MR717034
  8. [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. Zbl0655.35049MR932775
  9. [9] O. A. Ladyzhenskaja – V. A. Solonnikov – N. N. Uraltseva, “Linear and quasilinear equations of parabolic type”, Amer. Math. Soc. Providence, RI, 1968. 

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