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Keywords:
boundary value problem; nonlinear parabolic systems; solvability
boundary value problem; nonlinear parabolic systems; solvability
Summary:
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.
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[1] Amann H.: Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Rational Mech. Anal, 92 (1986), 2 153-192. MR 0816618 | Zbl 0596.35061
[2] Giaquinta M., Modica G.: Local existence for quasilinear parabolic systems under nonlinear boundary conditions. Ann. Mat. Pura Appl. (4) 149 (1987), 41-59. MR 0932775 | Zbl 0655.35049
[3] 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. MR 1668390 | Zbl 0953.35059
[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). MR 1746069
[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. MR 0826871 | Zbl 0595.58013
[7] Struwe M.: Variationals Methods. Applications to Partial Differential Equations and Hamiltonian Systems. Springer-Verlag, Berlin, 1990. MR 1078018
[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. MR 0717034 | Zbl 0516.49003
[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. MR 1800172 | Zbl 1046.35047
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