The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition

Volker Pluschke; Frank Weber

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 1, page 13-38
  • ISSN: 0010-2628

Abstract

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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 - ∂ u / ∂ ν A = g ( · , · , u ) with a locally defined, L r -bounded function g ( t , · , ξ ) . 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 ∞ , which is required by the local assumptions on g , is derived by a technique due to J. Moser.

How to cite

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Pluschke, Volker, and Weber, Frank. "The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition." Commentationes Mathematicae Universitatis Carolinae 40.1 (1999): 13-38. <http://eudml.org/doc/248386>.

@article{Pluschke1999,
abstract = {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 local assumptions on $g$, is derived by a technique due to J. Moser.},
author = {Pluschke, Volker, Weber, Frank},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method; parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method},
language = {eng},
number = {1},
pages = {13-38},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition},
url = {http://eudml.org/doc/248386},
volume = {40},
year = {1999},
}

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
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 1
SP - 13
EP - 38
AB - 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 local assumptions on $g$, is derived by a technique due to J. Moser.
LA - eng
KW - parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method; parabolic-elliptic problem; nonlinear Neumann boundary condition; Rothe method
UR - http://eudml.org/doc/248386
ER -

References

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