| Title:
|
The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition (English) |
| Author:
|
Pluschke, Volker |
| Author:
|
Weber, Frank |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
1 |
| Year:
|
1999 |
| Pages:
|
13-38 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
parabolic-elliptic problem |
| Keyword:
|
nonlinear Neumann boundary condition |
| Keyword:
|
Rothe method |
| MSC:
|
35K65 |
| MSC:
|
35M10 |
| MSC:
|
65N40 |
| idZBL:
|
Zbl 1060.35528 |
| idMR:
|
MR1715200 |
| . |
| Date available:
|
2009-01-08T18:49:28Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119061 |
| . |
| Reference:
|
[1] Browder F.E.: Estimates and existence theorems for elliptic boundary value problems.Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 365-372. Zbl 0093.29402, MR 0132913 |
| Reference:
|
[2] Filo J.: A nonlinear diffusion equation with nonlinear boundary conditions: Method of lines.Math. Slovaca 38.3 (1988), 273-296. Zbl 0664.35049, MR 0977906 |
| Reference:
|
[3] Fučik S., John O., Kufner A.: Function Spaces.Noordhoff International Publishing, Leyden, 1977. MR 0482102 |
| Reference:
|
[4] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order.Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Zbl 1042.35002, MR 0737190 |
| Reference:
|
[5] Jäger W., Kačur J.: Solution of porous medium type systems by linear approximation schemes.Numer. Math. 60 (1991), 407-427. MR 1137200 |
| Reference:
|
[6] Kačur J.: Method of Rothe in Evolution Equations.BSB Teubner Verlagsgesellschaft, Leipzig, 1985. MR 0834176 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 0111.09301, MR 0170091 |
| Reference:
|
[9] Pluschke V.: Local solution of parabolic equations with strongly increasing nonlinearity by the Rothe method.Czechoslovak Math. J. 38 , 113 (1988), 642-654. Zbl 0671.35037, MR 0962908 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[12] Pluschke V.: Rothe's method for semilinear parabolic problems with degeneration.Math. Nachr. 156 (1992), 283-295. Zbl 0794.35088, MR 1233950 |
| Reference:
|
[13] Rektorys K.: The Method of Discretization in Time and Partial Differential Equations.D.Reidel Publishing Company, Dordrecht, Boston, London, 1982. Zbl 0522.65059, MR 0689712 |
| Reference:
|
[14] Schechter M.: Coerciveness in $L_p$.Trans. Amer. Math. Soc. 107 (1963), 10-29. MR 0146690 |
| Reference:
|
[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 |
| Reference:
|
[16] Triebel H.: Interpolation Theory, Function Spaces, Differential Operators.VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Zbl 0830.46028, MR 0500580 |
| Reference:
|
[17] Weber F.: Die Anwendung der Rothe-Methode auf parabolische Probleme mit nichtlinearen Neumannschen Randbedingungen.Dissertation, Martin-Luther-Universität, Halle-Wittenberg, 1995. |
| Reference:
|
[18] Zeidler E.: Nonlinear Functional Analysis and its Applications - Volume II/B: Nonlinear Monotone Operators.Springer-Verlag, New York, Berlin, Heidelberg, 1990. |
| . |
