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Local Strong Solutions for the Non-Linear Thermoelastic Plate Equation on Rectangular Domains in $L^p$-Spaces

We consider the non-linear thermoelastic plate equation in rectangular domains $Ω$. More precisely, $Ω$ is considered to be given as the Cartesian product of whole or half spaces and a cube. First the linearized equation is treated as an abstract Cauchy problem in $L^p$-spaces. We take advantage of the structure of $Ω$ and apply operator-valued Fourier multiplier results to infer an $\mathcal R$-bounded $\mathcal H^\infty$-calculus. With the help of maximal $L^p$-regularity existence and uniqueness of local real-analytic strong solutions together with analytic dependency on the data is shown.