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Let $\Omega$ be a bounded domain. I am looking for techniques to show existence of solutions to dynamic boundary problems of the form $$\Delta u = 0 \quad\text{on}\quad \Omega \times (0,T)\\ \qquad\qquad\frac{du}{d\nu}= \frac{d}{dt}(f(u)) \quad\text{on}\quad \partial\Omega \times (0,T)\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!u(t=0) = u_0\quad\text{on}\quad \Omega$$ where $f$ is a (not necessarily globally Lipschitz) nonlinearity.

I know of two ways: 1) converting the problem to using the Dirichlet to Neumann map to a PDE on the boundary (as Lions does), and 2) utilising results from the theory of accretive monotone operators by discretizing the equation in time and then solving an elliptic problem etc.

I do not wish to use either of these. Are there any other methods?

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I would take a look at the paper by Ciprian Gal and Martin Meyries about elliptic problems with nonlinear time dependent boundary conditions. They treat a similar equation like the one you have and use maximal $L^p$-regularity for the linearized equation to show existence of solutions.

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