# Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition

### Arnd Rösch

Universität Duisburg-Essen, Germany

## Abstract

We consider the heat equation $\frac{\partial u}{\partial t} (t, x) = \Delta _x u(t,x)$ with a nonlinear function a in the boundary condition $\frac{\partial u}{\partial n} (t, x) =\alpha ((u(t,z))(\vartheta – u(t,x))$ depending on the boundary values $x$ of the solution u of the initial-boundary value problem only and belonging to a set of admissible differentiable or uniformly Lipschitz continuous functions. For this problem Lipschitz continuity and Fréchet differentiability of the mapping $\Phi : \alpha \mapsto x$ under different assumptions are derived.

## Cite this article

Arnd Rösch, Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition. Z. Anal. Anwend. 15 (1996), no. 3, pp. 603–618

DOI 10.4171/ZAA/718