# Article

Full entry | PDF   (0.3 MB)
Keywords:
boundary value problem; nonlinear parabolic systems; solvability
Summary:
A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval $[0,T)$ solution to the Cauchy-Neumann problem is studied. For the situation when the local energies'' of the solution are uniformly bounded on $[0,T)$, smooth extendibility of the solution up to $t=T$ is proved. In the case when $[0,T)$ defines the maximal interval of the existence of a smooth solution, the singular set at the moment $t=T$ is described.
References:
[1] 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
[2] Arkhipova A.: Local and global in time solvability of the Cauchy-Dirichlet problem to a class of nonlinear nondiagonal parabolic systems. Algebra & Analysis 11 6 (1999), 81-119 (Russian). MR 1746069
[3] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60 (1985), 558-581. MR 0826871 | Zbl 0595.58013
[4] Ladyzhenskaja O.A., Solonnikov V.A., Uraltseva N.N.: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Society, Providence, 1968.
[5] Giaquinta M., Modica G.: Local existence for quasilinear parabolic systems under non- linear boundary conditions. Ann. Mat. Pura Appl. 149 (1987), 41-59. MR 0932775
[6] Da Prato G.: Spazi ${\Cal L}^{p, \tau}(Ømega, \delta)$ e loro proprieta. Annali di Matem. LXIX (1965), 383-392. Zbl 0145.16207
[7] Campanato S.: Equazioni paraboliche del secondo ordine e spazi ${\Cal L}^{2, \delta} (Ømega, \delta)$. Ann. Mat. Pura Appl. 73 (1966), ser.4, 55-102. MR 0213737
[8] Arkhipova A.: On the Neumann problem for nonlinear elliptic systems with quadratic nonlinearity. St. Petersburg Math. J. 8 (1997), no. 5, 1-17; in Russian: Algebra & Analysis, St. Petersburg 8 (1996), no. 5. MR 1428990 | Zbl 0872.35020
[9] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud. 105, Princeton Univ. Press, Princeton, N.J., 1983. MR 0717034 | Zbl 0516.49003
[10] Nečas J., Šverák V.: On regularity of solutions of nonlinear parabolic systems. Ann. Scuola Norm. Sup. Pisa 18 ser. IV, F.1 (1991), 1-11. MR 1118218

Partner of