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

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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 $-\partial u/\partial \nu_A = g(\cdot,\cdot,u)$ 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 
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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/