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

Abstract

We consider the non-linear thermoelastic plate equation in rectangular domains $\Omega$. More precisely, $\Omega$ 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 $\Omega$ 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.

arXiv comment16 pages; final version

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• Maximal $L^p$-regularity and $H^{\infty }$-calculus for block operator matrices and applications