Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic nonlinearities II. Local and global solvability results

Arina A. Arkhipova

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 1, page 53-76
  • ISSN: 0010-2628

Abstract

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We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution is smooth almost everywhere and has an at most finite number of singular points.

How to cite

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Arkhipova, Arina A.. "Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic nonlinearities II. Local and global solvability results." Commentationes Mathematicae Universitatis Carolinae 42.1 (2001): 53-76. <http://eudml.org/doc/248781>.

@article{Arkhipova2001,
abstract = {We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution is smooth almost everywhere and has an at most finite number of singular points.},
author = {Arkhipova, Arina A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {boundary value problem; nonlinear parabolic systems; solvability; nonlinear boundary value problem; nonlinear parabolic systems},
language = {eng},
number = {1},
pages = {53-76},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic nonlinearities II. Local and global solvability results},
url = {http://eudml.org/doc/248781},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Arkhipova, Arina A.
TI - Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic nonlinearities II. Local and global solvability results
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 1
SP - 53
EP - 76
AB - We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution is smooth almost everywhere and has an at most finite number of singular points.
LA - eng
KW - boundary value problem; nonlinear parabolic systems; solvability; nonlinear boundary value problem; nonlinear parabolic systems
UR - http://eudml.org/doc/248781
ER -

References

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  4. Arkhipova A., Local and global in time solvability of the Cauchy-Dirichlet problem to a class of nonlinear nondiagonal parabolic systems, Algebra & Analysis 11 (1999), no. 6, 81-119 (Russian). MR1746069
  5. Ladyzhenskaja O.A., Solonnikov V.A., Uraltseva N.N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Society, Providence, 1968. 
  6. Struwe M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558-581. (1985) Zbl0595.58013MR0826871
  7. Struwe M., Variationals Methods. Applications to Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. MR1078018
  8. Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud. 105, Princeton Univ. Press, Princeton, NJ, 1983. Zbl0516.49003MR0717034
  9. Arkhipova A., Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities. I. On the continuability of smooth solutions, Comment. Math. Univ. Carolinae 41 (2000), 4 693-718. (2000) Zbl1046.35047MR1800172

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