The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition
Pluschke, Volker ; Weber, Frank
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999), p. 13-38 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1999-01-01
Classification:  35K65,  35M10,  65N40 
@article{119061,
     author = {Volker Pluschke and Frank Weber},
     title = {The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {40},
     year = {1999},
     pages = {13-38},
     zbl = {1060.35528},
     mrnumber = {1715200},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119061}
}

Pluschke, Volker; Weber, Frank. The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) pp. 13-38. http://gdmltest.u-ga.fr/item/119061/

Browder F.E. Estimates and existence theorems for elliptic boundary value problems, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 365-372. (1959) | MR 0132913 | Zbl 0093.29402

Filo J. A nonlinear diffusion equation with nonlinear boundary conditions: Method of lines, Math. Slovaca 38.3 (1988), 273-296. (1988) | MR 0977906 | Zbl 0664.35049

Fučik S.; John O.; Kufner A. Function Spaces, Noordhoff International Publishing, Leyden, 1977. | MR 0482102

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

Jäger W.; Kačur J. Solution of porous medium type systems by linear approximation schemes, Numer. Math. 60 (1991), 407-427. (1991) | MR 1137200

Kačur J. Method of Rothe in Evolution Equations, BSB Teubner Verlagsgesellschaft, Leipzig, 1985. | MR 0834176

Kačur J. On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev-spaces I & II, Math. Z. 203 (1990), 153-171, 569-579. (1990) | MR 1030713

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. (1960) | MR 0170091 | Zbl 0111.09301

Pluschke V. Local solution of parabolic equations with strongly increasing nonlinearity by the Rothe method, Czechoslovak Math. J. 38 , 113 (1988), 642-654. (1988) | MR 0962908 | Zbl 0671.35037

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

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

Pluschke V. Rothe's method for semilinear parabolic problems with degeneration, Math. Nachr. 156 (1992), 283-295. (1992) | MR 1233950 | Zbl 0794.35088

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

Schechter M. Coerciveness in $L_p$, Trans. Amer. Math. Soc. 107 (1963), 10-29. (1963) | MR 0146690

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. (1989) | MR 1026508

Triebel H. Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. | MR 0500580 | Zbl 0830.46028

Weber F. Die Anwendung der Rothe-Methode auf parabolische Probleme mit nichtlinearen Neumannschen Randbedingungen, Dissertation, Martin-Luther-Universität, Halle-Wittenberg, 1995.

Zeidler E. Nonlinear Functional Analysis and its Applications - Volume II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, Berlin, Heidelberg, 1990.