# Article

Full entry | PDF   (0.3 MB)
Keywords:
boundary value problems; nonlinear parabolic systems; solvability
Summary:
A class of $q$-nonlinear parabolic systems with a nondiagonal principal matrix and strong nonlinearities in the gradient is considered.We discuss the global in time solvability results of the classical initial boundary value problems in the case of two spatial variables. The systems with nonlinearities $q\in (1,2)$, $q=2$, $q>2$, are analyzed.
References:
[1] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N.: Linear and Quasilinear Equations of Parabolic Type. Amer. Math Society, Providence, 1968.
[2] Stará J., John O.: Some (new) counterexamples of parabolic systems. Comment. Math. Univ. Carolin. 36 (1995), 503–510. MR 1364491
[3] Chen Y., Struwe M.: Existence and partial regularity results for the heatflow for harmonic maps. Math. Z. 201 (1989), 83–103. MR 0990191
[4] Chang K.-C.: Heat flow and boundary value problem for harmonic maps. Ann. Inst. Henri Poincare 6 (1989), 363–395. MR 1030856 | Zbl 0687.58004
[5] 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., St. Petersburg Univ. 16 (1997), 3–40. MR 1668390 | Zbl 0953.35059
[6] Arkhipova A.: Local and global solvability of the Cauchy-Dirichlet problem for a class of nonlinear nondiagonal parabolic systems. St. Petersburg Math. J. 11 (2000), 989–1017. MR 1746069 | Zbl 0973.35095
[7] 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. Carolin. 41 (2000), 693–718. MR 1800172 | Zbl 1046.35047
[8] Arkhipova A.: Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities. II. Local and global solvability results. Comment. Math. Univ. Carolin. 42 (2001), 53–76. MR 1825372

Partner of