Abstract
The Cauchy-Dirichlet problem for quasilinear parabolic systems of second-order equations is considered in the case of two spatial variables. Under the condition that the corresponding elliptic operator has variational structure, the global in time solvability is established. The solution is smooth almost everywhere and the number of singular points is finite. Sufficient conditions that guarantee the absence of singular points are given. Bibliography: 23 titles.
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References
M. Giaquinta and G. Modica, “Local existence for quasilinear parabolic systems under nonlinear boundary conditions”,Ann. Mat. Pura Appl., Ser., 4,149, 41–59 (1987).
S. Campanato, “Equazioni paraboliche del secondo ordine e spaziL 2θ (ω, δ),”Ann. Mat. Pura Appl., Ser. 4,73, 55–102 (1966).
P. Tolksdorf, “On some parabolic variational problems with quadratic growth,”Ann. Scuola Norm. Sup. Pisa, Ser. 4,13, No. 2, 193–223 (1986).
W. Wieser, “On parabolic systems with variational structure,”Manuscripta Math.,54, No. 1-2, 53–82 (1985).
M. Struwe, “On the evolution of harmonic mappings of Riemannian surfaces,”Comment. Math. Helv.,60, No. 4, 558–581 (1985).
Y. Chen and M. Struwe, “Existence and partial regularity results for the heat flow for harmonic maps,”Math. Z.,201, No. 1, 83–103 (1989).
K. C. Chang, “Heat flow and boundary value problem for harmonic maps,”Ann. Inst. H. Poincaré Anal. Non Linéaire,6, No. 5, 363–395 (1989).
M. Struwe, “On the evolution of harmonic maps in higher dimensions,”J. Diff. Geom. 28, No. 3, 485–502 (1988).
M. Struwe,Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer Verlag, Berlin (1990).
M. Giaquinta and M. Struwe, “On the partial regularity of weak solutions of nonlinear parabolic systems,”Math. Z.,179, 437–451 (1982).
M. Marino and A. Maugeri, “Partial Hölder continuity of solutions of nonlinear parabolic systems of second order with quadratic growth,”Boll. Un. Mat. Ital. B.,3, No. 2, 397–435 (1989).
M. Marino and A. Maugeri, “Maximum principle for parabolic systems”,St. Petersburg Math. J.,3, No. 6, 1351–1358 (1992).
M. Marino and A. Maugeri, “Regularity for parabolic systems in divergence form,”Methods Real Anal. PDE, Quaderno n. 14, Napoli (1992), pp. 63–82.
M. Marino and A. Maugeri, “Boundedness results for parabolic systems,”Methods Real Anal. PDE, Quaderno n. 14, Napoli (1992), pp. 39–62.
M. Marino and A. Maugeri, “L 2,λ Regularity of the spatial derivatives of the solutions to parabolic systems in divergence form,”Ann. Mat. Pura Appl. Ser. 4,164, 275–298 (1993).
M. Marino and A. Maugeri, “A remark on the note: ‘Partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth’,”Rend. Sem. Mat. Univ. Padova,95, 23–28 (1996).
J. Naumann and J. Wolf, “Interior differentiability of the weak solutions to parabolic systems with quadratic growth nonlinearities,” 1996.
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva,Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence (1968).
M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton (1983).
O. A. Ladyzhenskaia,The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969).
J. Nečas and V. Šverak, “On regularity of solutions of nonlinear parabolic systems,”Ann. Scuola Norm. Sup. Pisa, Ser. 4,18, No. 1, 1–11 (1991).
C. Morrey,Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York (1966).
A. A. Arkhipova, “On the regularity of solutions of boundary-value problems for quasilinear elliptic systems with quadratic nonlinearity,”J. Math. Sci. 80, No. 6, 2208–2225 (1996).
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Translated fromProblemy Matematicheskogo Analiza No. 16, 1997, pp. 3–40.
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Arkhipova, A.A. 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, 4231–4255 (1998). https://doi.org/10.1007/BF02433433
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DOI: https://doi.org/10.1007/BF02433433