Volker Pluschke, Frank Weber 6DA ?= IKJE B = F=H=>E?AEFJE? AGK=JE MEJD = EA=H AK= >K@=HO ?@EJE

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Volker Pluschke, Frank Weber

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

Comment.Math.Univ.Carolinae 40,1 (1999) 13-38.

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

−∂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 inL∞, which is required by the local assumptions ong, is derived by a technique due to J. Moser.

Keywords: parabolic-elliptic problem, nonlinear Neumann boundary condition, Rothe method

AMS Subject Classification: 35K65, 65N40, 35M10

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Volker Pluschke, Frank Weber

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

Volker Pluschke, Frank Weber 6DA ?= IKJE B = F=H=>E?AEFJE? AGK=JE MEJD = EA=H AK= >K@=HO ?@EJE