Geodesic
The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 1, pp. 13-38.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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.
Classification : 35K65, 35M10, 65N40
Keywords: parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method 
@article{CMUC_1999__40_1_a2,
     author = {Pluschke, Volker and Weber, Frank},
     title = {The local solution of a parabolic-elliptic equation with a nonlinear {Neumann} boundary condition},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {13--38},
     publisher = {mathdoc},
     volume = {40},
     number = {1},
     year = {1999},
     mrnumber = {1715200},
     zbl = {1060.35528},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1999__40_1_a2/}
}
TY  - JOUR
AU  - Pluschke, Volker
AU  - Weber, Frank
TI  - The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1999
SP  - 13
EP  - 38
VL  - 40
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMUC_1999__40_1_a2/
LA  - en
ID  - CMUC_1999__40_1_a2
ER  - 
%0 Journal Article
%A Pluschke, Volker
%A Weber, Frank
%T The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition
%J Commentationes Mathematicae Universitatis Carolinae
%D 1999
%P 13-38
%V 40
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMUC_1999__40_1_a2/
%G en
%F CMUC_1999__40_1_a2
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) no. 1, pp. 13-38. http://geodesic.mathdoc.fr/item/CMUC_1999__40_1_a2/