Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method
parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method
Summary:
We investigate a parabolic-elliptic problem, where the time derivative is multiplied by a coefficient which may vanish on time-dependent spatial subdomains. The linear equation is supplemented by a nonlinear Neumann boundary condition $-\partial u/\partial \nu_A = g(\cdot,\cdot,u)$ with a locally defined, $L_r$-bounded function $g(t,\cdot,\xi)$. We prove the existence of a local weak solution to the problem by means of the Rothe method. A uniform a priori estimate for the Rothe approximations in $L_{\infty}$, which is required by the {\it local} assumptions on $g$, is derived by a technique due to J. Moser.
References:
[1] Browder F.E.: Estimates and existence theorems for elliptic boundary value problems. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 365-372. MR 0132913 | Zbl 0093.29402
[2] Filo J.: A nonlinear diffusion equation with nonlinear boundary conditions: Method of lines. Math. Slovaca 38.3 (1988), 273-296. MR 0977906 | Zbl 0664.35049
[3] Fučik S., John O., Kufner A.: Function Spaces. Noordhoff International Publishing, Leyden, 1977. MR 0482102
[4] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 0737190 | Zbl 1042.35002
[5] Jäger W., Kačur J.: Solution of porous medium type systems by linear approximation schemes. Numer. Math. 60 (1991), 407-427. MR 1137200
[6] Kačur J.: Method of Rothe in Evolution Equations. BSB Teubner Verlagsgesellschaft, Leipzig, 1985. MR 0834176
[7] Kačur J.: On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev-spaces I & II. Math. Z. 203 (1990), 153-171, 569-579. MR 1030713
[8] Moser J.: A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13.3 (1960), 457-468. MR 0170091 | Zbl 0111.09301
[9] Pluschke V.: Local solution of parabolic equations with strongly increasing nonlinearity by the Rothe method. Czechoslovak Math. J. 38 , 113 (1988), 642-654. MR 0962908 | Zbl 0671.35037
[10] Pluschke V.: Construction of bounded solutions to degenerated parabolic equations by the Rothe method. in: {Complex Analysis and Differential Equations} (in Russian), Bashkirian State University, Ufa, 1990, pp.58-75. MR 1252652
[11] Pluschke V.: $L_{\infty} $-Estimates and Uniform Convergence of Rothe's Method for Quasilinear Parabolic Differential Equations. in: R. Kleinmann, R. Kress, E. Martensen, {Direct and Inverse Boundary Value Problems}, Verlag Peter Lang GmbH, Frankfurt am Main, 1991, pp.187-199. MR 1215747
[12] Pluschke V.: Rothe's method for semilinear parabolic problems with degeneration. Math. Nachr. 156 (1992), 283-295. MR 1233950 | Zbl 0794.35088
[13] Rektorys K.: The Method of Discretization in Time and Partial Differential Equations. D.Reidel Publishing Company, Dordrecht, Boston, London, 1982. MR 0689712 | Zbl 0522.65059
[14] Schechter M.: Coerciveness in $L_p$. Trans. Amer. Math. Soc. 107 (1963), 10-29. MR 0146690
[15] Slodička M.: Error estimate of approximate solution for a quasilinear parabolic integrodifferential equation in the $L_p$-space. Aplikace Matematiky 34 6 (1989), 439-448. MR 1026508
[16] Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 0500580 | Zbl 0830.46028
[17] Weber F.: Die Anwendung der Rothe-Methode auf parabolische Probleme mit nichtlinearen Neumannschen Randbedingungen. Dissertation, Martin-Luther-Universität, Halle-Wittenberg, 1995.
[18] Zeidler E.: Nonlinear Functional Analysis and its Applications - Volume II/B: Nonlinear Monotone Operators. Springer-Verlag, New York, Berlin, Heidelberg, 1990.
Search
Partner of
